Variant 2: Higher-Order / Formulas-as-Types Proof Assistant (RoqC / Dependent NBG)

1. Propositional & Constructive Logic (NBG-001–NBG-019)

Constructive lambda-term proofs of propositional axiom schemas and foundational combinators.

Num NBG Formulas-as-Types Record (Vertical Field Table)

Identifier	`thm_A01`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), (A → (B → A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (a :A ). (λ (b :B ). a )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (_ : B) -> A))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-002

Identifier	`thm_A02`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ∀(C :Prop ), ((A → (B → C )) → ((A → B ) → (A → C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (C :Prop ). (λ (f :(A → (B → C ))). (λ (g :(A → B )). (λ (x :A ). ((f x ) (g x )))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> (Pi (_ : B) -> C))) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : A) -> C))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-003

Identifier	`thm_A03`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A ∧ B ) → A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (p :(A ∧ B )). fst (p )))) ≡ (λ (A :Prop ). (λ (B :Prop ). fst )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/definitional-eq.html}{\equiv} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> A)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check; both expanded & η-reduced fst forms verified equal via refl)

NBG-004

Identifier	`thm_A04`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A ∧ B ) → B ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (p :(A ∧ B )). snd (p )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> B)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-005

Identifier	`thm_A05`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), (A → (B → (A ∧ B ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (a :A ). (λ (b :B ). ⟨a , b ⟩)))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (_ : B) -> (Sigma (_ : A) * B)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-006

Identifier	`thm_A06`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), (A → ∀(C :Prop ), ((A → C ) → ((B → C ) → C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (a :A ). (λ (C :Prop ). (λ (fa :(A → C )). (λ (fb :(B → C )). (fa a ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> C)) -> (Pi (_ : (Pi (_ : B) -> C)) -> C))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-007

Identifier	`thm_A07`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), (B → ∀(C :Prop ), ((A → C ) → ((B → C ) → C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (b :B ). (λ (C :Prop ). (λ (fa :(A → C )). (λ (fb :(B → C )). (fb b ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : B) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> C)) -> (Pi (_ : (Pi (_ : B) -> C)) -> C))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-008

Identifier	`thm_A08`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ∀(C :Prop ), ((A → C ) → ((B → C ) → ((∀(D :Prop ), ((A → D ) → ((B → D ) → D ))) → C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{D} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (C :Prop ). (λ (fa :(A → C )). (λ (fb :(B → C )). (λ (disj :∀(D :Prop ), ((A → D ) → ((B → D ) → D ))). (((disj C ) fa ) fb ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{disj}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{D} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{disj}} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> C)) -> (Pi (_ : (Pi (_ : B) -> C)) -> (Pi (_ : (Pi (D : Prop) -> (Pi (_ : (Pi (_ : A) -> D)) -> (Pi (_ : (Pi (_ : B) -> D)) -> D)))) -> C))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-009

Identifier	`thm_A09`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A → B ) → ((A → ¬B ) → ¬A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (f :(A → B )). (λ (g :(A → ¬B )). (λ (a :A ). ((g a ) (f a ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : A) -> (Pi (_ : B) -> False))) -> (Pi (_ : A) -> False)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-010

Identifier	`thm_A10`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), (¬¬A → A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (dneg :¬¬A ). ((ax_dneg_elim A ) dneg ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{dneg}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_dneg\_elim}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{dneg}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (_ : (Pi (_ : (Pi (_ : A) -> False)) -> False)) -> A))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-011

Identifier	`thm_A11`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), (⊥ → A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/bot-const.html}{\bot} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (falsum :⊥). ((ax_false_elim A ) falsum ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{falsum}} \href{symbols/typing-colon.html}{:} \href{symbols/bot-const.html}{\bot} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_false\_elim}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{falsum}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (_ : False) -> A))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-012

Identifier	`thm_A12`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), (((A → B ) → A ) → A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (h :((A → B ) → A )). (((ax_peirce A ) B ) h )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_peirce}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : (Pi (_ : A) -> B)) -> A)) -> A)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-013

Identifier	`thm_identity`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), (A → A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (x :A ). x )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (_ : A) -> A))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-014

Identifier	`thm_modus_ponens`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A → B ) → (A → B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (f :(A → B )). (λ (a :A ). (f a ))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : A) -> B))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-015

Identifier	`thm_transitivity`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ∀(C :Prop ), ((A → B ) → ((B → C ) → (A → C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (C :Prop ). (λ (f :(A → B )). (λ (g :(B → C )). (λ (a :A ). (g (f a )))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{g} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : B) -> C)) -> (Pi (_ : A) -> C))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-016

Identifier	`thm_and_comm`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A ∧ B ) → (B ∧ A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (p :(A ∧ B )). ⟨snd (p ), fst (p )⟩))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/and-conn.html}{\land} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{p} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> (Sigma (_ : B) * A))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-017

Identifier	`thm_contraposition`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :Prop ), ((A → B ) → (¬B → ¬A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{B} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :Prop ). (λ (f :(A → B )). (λ (nb :¬B ). (λ (a :A ). (nb (f a ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nb}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nb}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : B) -> False)) -> (Pi (_ : A) -> False)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-018

Identifier	`thm_D01`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Prop ), ∀(B :(Class → Prop )), ((∀(x :Class ), (A → (B x ))) → (A → ∀(x :Class ), (B x ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Prop ). (λ (B :(Class → Prop )). (λ (h :∀(x :Class ), (A → (B x ))). (λ (a :A ). (λ (x :Class ). ((h x ) a )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Prop) -> (Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (_ : (Pi (x : Class) -> (Pi (_ : A) -> (B x)))) -> (Pi (_ : A) -> (Pi (x : Class) -> (B x))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-019

Identifier	`thm_forall_inst`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(B :(Class → Prop )), ∀(c :Class ), ((∀(x :Class ), (B x )) → (B c )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{c} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{c} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (B :(Class → Prop )). (λ (c :Class ). (λ (h :∀(x :Class ), (B x )). (h c )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{c} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/variables.html}{c} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (c : Class) -> (Pi (_ : (Pi (x : Class) -> (B x))) -> (B c))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

2. First-Order Quantification & Identity Types (NBG-020–NBG-030)

Dependent product (Pi), dependent sum (Sigma), and Martin-Löf identity type theorems.

Num NBG Formulas-as-Types Record (Vertical Field Table)

NBG-020

Identifier	`thm_exists_intro`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(B :(Class → Prop )), ∀(c :Class ), ((B c ) → ∃(x :Class ), (B x )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{c} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{c} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (B :(Class → Prop )). (λ (c :Class ). (λ (witness :(B c )). ⟨c , witness ⟩))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{c} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{witness}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/variables.html}{c} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/variables.html}{c} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{witness}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (c : Class) -> (Pi (_ : (B c)) -> (Sigma (x : Class) * (B x)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-021

Identifier	`def_isSet`
Classification	Conservative Definition
Statement (Type)	0∀(X :Class ), ((set (X ) → ∃(Y :Class ), (X ∈ Y )) ∧ ((∃(Y :Class ), (X ∈ Y )) → set (X ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_isSet_def 0\(\href{symbols/variables.html}{\mathsf{plan\_isSet\_def}} \)
SML Kernel AST	`(Pi (X : Class) -> (Sigma (_ : (Pi (_ : (Id Prop (Set X) true)) -> (Sigma (Y : Class) * (Id Prop ((In X) Y) true)))) * (Pi (_ : (Sigma (Y : Class) * (Id Prop ((In X) Y) true))) -> (Id Prop (Set X) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-022

Identifier	`def_isProper`
Classification	Conservative Definition
Statement (Type)	0∀(K :Class ), ((proper (K ) → ¬set (K )) ∧ (¬set (K ) → proper (K ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/proper-pred.html}{\mathsf{proper}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/neg-conn.html}{\neg} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/proper-pred.html}{\mathsf{proper}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_isProper_def 0\(\href{symbols/variables.html}{\mathsf{plan\_isProper\_def}} \)
SML Kernel AST	`(Pi (K : Class) -> (Sigma (_ : (Pi (_ : (Id Prop (Proper K) true)) -> (Pi (_ : (Id Prop (Set K) true)) -> False))) * (Pi (_ : (Pi (_ : (Id Prop (Set K) true)) -> False)) -> (Id Prop (Proper K) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-023

Identifier	`def_empty_set`
Classification	Conservative Definition
Statement (Type)	0∀(X :Class ), (((X = ∅) → ∀(z :Class ), ¬(z ∈ X )) ∧ ((∀(z :Class ), ¬(z ∈ X )) → (X = ∅))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_empty_set_def 0\(\href{symbols/variables.html}{\mathsf{plan\_empty\_set\_def}} \)
SML Kernel AST	`(Pi (X : Class) -> (Sigma (_ : (Pi (_ : (Id Class X EmptySet)) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> False)))) * (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> False))) -> (Id Class X EmptySet))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-024

Identifier	`def_universal_class`
Classification	Conservative Definition
Statement (Type)	0∀(X :Class ), (((X = V ) → ∀(z :Class ), (((z ∈ X ) → set (z )) ∧ (set (z ) → (z ∈ X )))) ∧ ((∀(z :Class ), (((z ∈ X ) → set (z )) ∧ (set (z ) → (z ∈ X )))) → (X = V ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_universal_class_def 0\(\href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \)
SML Kernel AST	`(Pi (X : Class) -> (Sigma (_ : (Pi (_ : (Id Class X V)) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop (Set z) true))) * (Pi (_ : (Id Prop (Set z) true)) -> (Id Prop ((In z) X) true)))))) * (Pi (_ : (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop (Set z) true))) * (Pi (_ : (Id Prop (Set z) true)) -> (Id Prop ((In z) X) true))))) -> (Id Class X V))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-025

Identifier	`def_subclass`
Classification	Conservative Definition
Statement (Type)	0∀(X :Class ), ∀(Y :Class ), (((X ⊆ Y ) → ∀(z :Class ), ((z ∈ X ) → (z ∈ Y ))) ∧ ((∀(z :Class ), ((z ∈ X ) → (z ∈ Y ))) → (X ⊆ Y ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_subclass_def 0\(\href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (Y : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((Subset X) Y) true)) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) Y) true))))) * (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) Y) true)))) -> (Id Prop ((Subset X) Y) true)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-026

Identifier	`thm_emptyset_is_subclass`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), ∀(z :Class ), ((z ∈ ∅) → (z ∈ X )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (λ (z :Class ). (λ (h :(z ∈ ∅)). ((ax_false_elim (z ∈ X )) ((ax_empty_no_elements z ) h ))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_false\_elim}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_empty\_no\_elements}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) EmptySet) true)) -> (Id Prop ((In z) X) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-027

Identifier	`ax_extensionality`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(X :Class ), ∀(Y :Class ), ((∀(z :Class ), (((z ∈ X ) → (z ∈ Y )) ∧ ((z ∈ Y ) → (z ∈ X )))) → (X = Y )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_extensionality 0\(\href{symbols/variables.html}{\mathsf{plan\_extensionality}} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (Y : Class) -> (Pi (_ : (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) Y) true))) * (Pi (_ : (Id Prop ((In z) Y) true)) -> (Id Prop ((In z) X) true))))) -> (Id Class X Y))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-028

Identifier	`ax_heredity`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (((A ⊆ B ) ∧ set (B )) → set (A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_heredity 0\(\href{symbols/variables.html}{\mathsf{plan\_heredity}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((Subset A) B) true)) * (Id Prop (Set B) true))) -> (Id Prop (Set A) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-029

Identifier	`ax_exponent`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(x :Class ), (set (x ) → set (𝒫 (x ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/power-op.html}{\mathcal{P}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_exponent 0\(\href{symbols/variables.html}{\mathsf{plan\_exponent}} \)
SML Kernel AST	`(Pi (x : Class) -> (Pi (_ : (Id Prop (Set x) true)) -> (Id Prop (Set (Power x)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-030

Identifier	`ax_regularity`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(X :Class ), ((𝒫 (X ) ⊆ X ) → (V ⊆ X )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/power-op.html}{\mathcal{P}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_regularity 0\(\href{symbols/variables.html}{\mathsf{plan\_regularity}} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (_ : (Id Prop ((Subset (Power X)) X) true)) -> (Id Prop ((Subset V) X) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

3. NBG/MK Set & Class Foundations, Ordinals & Dunaev Pairs (NBG-031–NBG-064)

Sethood, proper classes, Russell's paradox, ordinals, omega, Dunaev tagged pairs, cardinals, and Tarski universes.

Num NBG Formulas-as-Types Record (Vertical Field Table)

NBG-031

Identifier	`def_ord`
Classification	Conservative Definition
Statement (Type)	0∀(A :Class ), ((Ord (A ) → (Tr (A ) ∧ (∀(B :Class ), ((B ∈ A ) → Tr (B ))))) ∧ ((Tr (A ) ∧ (∀(B :Class ), ((B ∈ A ) → Tr (B )))) → Ord (A ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/ord-pred.html}{\mathrm{Ord}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/trans-pred.html}{\mathrm{Tr}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/trans-pred.html}{\mathrm{Tr}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/trans-pred.html}{\mathrm{Tr}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/trans-pred.html}{\mathrm{Tr}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/ord-pred.html}{\mathrm{Ord}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_ord_def 0\(\href{symbols/variables.html}{\mathsf{plan\_ord\_def}} \)
SML Kernel AST	`(Pi (A : Class) -> (Sigma (_ : (Pi (_ : (Id Prop (Ord A) true)) -> (Sigma (_ : (Id Prop (Trans A) true)) * (Pi (B : Class) -> (Pi (_ : (Id Prop ((In B) A) true)) -> (Id Prop (Trans B) true)))))) * (Pi (_ : (Sigma (_ : (Id Prop (Trans A) true)) * (Pi (B : Class) -> (Pi (_ : (Id Prop ((In B) A) true)) -> (Id Prop (Trans B) true))))) -> (Id Prop (Ord A) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-032

Identifier	`ax_global_choice`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(A :Class ), ((∃(x :Class ), (x ∈ A )) → (choice (A ) ∈ A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/eps-binder.html}{\mathsf{choice}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_global_choice 0\(\href{symbols/variables.html}{\mathsf{plan\_global\_choice}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (_ : (Sigma (x : Class) * (Id Prop ((In x) A) true))) -> (Id Prop ((In (choice A)) A) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-033

Identifier	`thm_subset_refl`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), ∀(z :Class ), ((z ∈ X ) → (z ∈ X )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (λ (z :Class ). (λ (hz :(z ∈ X )). hz ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) X) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-034

Identifier	`thm_subset_trans`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), ∀(Y :Class ), ∀(Z :Class ), ((∀(z :Class ), ((z ∈ X ) → (z ∈ Y ))) → ((∀(z :Class ), ((z ∈ Y ) → (z ∈ Z ))) → ∀(z :Class ), ((z ∈ X ) → (z ∈ Z )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (λ (Y :Class ). (λ (Z :Class ). (λ (h1 :∀(z :Class ), ((z ∈ X ) → (z ∈ Y ))). (λ (h2 :∀(z :Class ), ((z ∈ Y ) → (z ∈ Z ))). (λ (z :Class ). (λ (hz :(z ∈ X )). ((h2 z ) ((h1 z ) hz ))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{h1}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{h2}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{h2}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{h1}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (Y : Class) -> (Pi (Z : Class) -> (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) Y) true)))) -> (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) Y) true)) -> (Id Prop ((In z) Z) true)))) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) Z) true))))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-035

Identifier	`def_dunaev_opair`
Classification	Conservative Definition
Statement (Type)	0∀(a :Class ), ∀(b :Class ), ((a , b ) = (a , b )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0def_dunaev_opair_witness 0\(\href{symbols/variables.html}{\mathsf{def\_dunaev\_opair\_witness}} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Id Class ((Pair a) b) ((DunaevPair a) b))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-036

Identifier	`def_relational_func`
Classification	Conservative Definition
Statement (Type)	0∀(F :Class ), ((function (F ) → ∀(x :Class ), ∀(y :Class ), ∀(z :Class ), ((((x , y ) ∈ F ) ∧ ((x , z ) ∈ F )) → (y = z ))) ∧ ((∀(x :Class ), ∀(y :Class ), ∀(z :Class ), ((((x , y ) ∈ F ) ∧ ((x , z ) ∈ F )) → (y = z ))) → function (F ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/func-pred.html}{\mathsf{function}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/func-pred.html}{\mathsf{function}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_relational_func_def 0\(\href{symbols/variables.html}{\mathsf{plan\_relational\_func\_def}} \)
SML Kernel AST	`(Pi (F : Class) -> (Sigma (_ : (Pi (_ : (Id Prop (Func F) true)) -> (Pi (x : Class) -> (Pi (y : Class) -> (Pi (z : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((In ((DunaevPair x) y)) F) true)) * (Id Prop ((In ((DunaevPair x) z)) F) true))) -> (Id Class y z))))))) * (Pi (_ : (Pi (x : Class) -> (Pi (y : Class) -> (Pi (z : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((In ((DunaevPair x) y)) F) true)) * (Id Prop ((In ((DunaevPair x) z)) F) true))) -> (Id Class y z)))))) -> (Id Prop (Func F) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-037

Identifier	`thm_zero_is_nat`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0(0 = 0 ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/numeral-0.html}{0} \href{symbols/eq-pred.html}{=} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0refl (0 ) 0\(\href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Nat 0 0)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-038

Identifier	`thm_succ_preserves_nat`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(n :Nat ), (succ (n ) = succ (n )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (n :Nat ). refl (succ (n ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (n : Nat) -> (Id Nat (succ n) (succ n)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-039

Identifier	`thm_zero_not_succ`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(n :Nat ), ∀(h :(succ (n ) = 0 )), ⊥ 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/bot-const.html}{\bot} \)
Proof Program	0(λ (n :Nat ). (λ (h :(succ (n ) = 0 )). ((ax_peano_zero_not_succ n ) h ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_peano\_zero\_not\_succ}} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (n : Nat) -> (Pi (h : (Id Nat (succ n) 0)) -> False))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-040

Identifier	`thm_succ_injective`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(n :Nat ), ∀(m :Nat ), ∀(h :(succ (n ) = succ (m ))), (n = m ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{m} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{m} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{m} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (n :Nat ). (λ (m :Nat ). (λ (h :(succ (n ) = succ (m ))). (((ax_peano_succ_inj n ) m ) h )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{m} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{m} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_peano\_succ\_inj}} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{m} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (n : Nat) -> (Pi (m : Nat) -> (Pi (h : (Id Nat (succ n) (succ m))) -> (Id Nat n m))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-041

Identifier	`thm_add_zero_left`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(n :Nat ), (n = n ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (n :Nat ). refl (n )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (n : Nat) -> (Id Nat n n))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-042

Identifier	`thm_mathematical_induction`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(P :(Nat → Prop )), ∀(base :(P 0 )), ∀(step :∀(n :Nat ), ∀(ih :(P n )), (P succ (n ))), ∀(n :Nat ), (P n ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{base}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{step}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ih}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (P :(Nat → Prop )). (λ (base :(P 0 )). (λ (step :∀(n :Nat ), ∀(ih :(P n )), (P succ (n ))). (λ (n :Nat ). nat_rec (P , base , step , n ))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{base}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/numeral-0.html}{0} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{step}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ih}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/omega-const.html}{\mathsf{succ}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{n} \href{symbols/typing-colon.html}{:} \href{symbols/omega-const.html}{\mathsf{Nat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/omega-const.html}{\mathsf{nat\_rec}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{P} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{base}} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{step}} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{n} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (P : (Pi (_ : Nat) -> Prop)) -> (Pi (base : (P 0)) -> (Pi (step : (Pi (n : Nat) -> (Pi (ih : (P n)) -> (P (succ n))))) -> (Pi (n : Nat) -> (P n)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-043

Identifier	`thm_v_ind`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Ind (V ) 0\(\href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0((ax_ind_intro V ) ⟨(snd (((fst ((plan_universal_class_def V )) refl (V )) ∅)) ax_empty_set ), (λ (y :Class ). (λ (hyInV :(y ∈ V )). (snd (((fst ((plan_universal_class_def V )) refl (V )) SuccOrd (y ))) ((ax_ord_succ_set y ) (fst (((fst ((plan_universal_class_def V )) refl (V )) y )) hyInV )))))⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_intro}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{ax\_empty\_set}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hyInV}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ord\_succ\_set}} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hyInV}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop (Ind V) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-044

Identifier	`thm_omega_def`
Classification	Conservative Definition
Statement (Type)	0∀(x :Class ), (((x ∈ ω ) → ∀(Q :Class ), (Ind (Q ) → (x ∈ Q ))) ∧ ((∀(Q :Class ), (Ind (Q ) → (x ∈ Q ))) → (x ∈ ω ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_omega_def 0\(\href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \)
SML Kernel AST	`(Pi (x : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In x) omega) true)) -> (Pi (Q : Class) -> (Pi (_ : (Id Prop (Ind Q) true)) -> (Id Prop ((In x) Q) true))))) * (Pi (_ : (Pi (Q : Class) -> (Pi (_ : (Id Prop (Ind Q) true)) -> (Id Prop ((In x) Q) true)))) -> (Id Prop ((In x) omega) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-045

Identifier	`thm_omega_set`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0set (ω ) 0\(\href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_omega_set 0\(\href{symbols/variables.html}{\mathsf{plan\_omega\_set}} \)
SML Kernel AST	`(Id Prop (Set omega) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-046

Identifier	`thm_omega_ind`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Ind (ω ) 0\(\href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0((ax_ind_intro ω ) ⟨(snd ((plan_omega_def ∅)) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). fst (((ax_ind_elim Q ) hIndQ ))))), (λ (y :Class ). (λ (hyInOmega :(y ∈ ω )). (snd ((plan_omega_def SuccOrd (y ))) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). ((snd (((ax_ind_elim Q ) hIndQ )) y ) (((fst ((plan_omega_def y )) hyInOmega ) Q ) hIndQ )))))))⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_intro}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop (Ind omega) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-047

Identifier	`thm_tr_omega`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Tr (ω ) 0\(\href{symbols/trans-pred.html}{\mathrm{Tr}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0((ax_omega_trans_from_ind ((ax_ind_intro ω ) ⟨(snd ((plan_omega_def ∅)) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). fst (((ax_ind_elim Q ) hIndQ ))))), (λ (y :Class ). (λ (hyInOmega :(y ∈ ω )). (snd ((plan_omega_def SuccOrd (y ))) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). ((snd (((ax_ind_elim Q ) hIndQ )) y ) (((fst ((plan_omega_def y )) hyInOmega ) Q ) hIndQ )))))))⟩)) ax_omega_elements_trans ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_omega\_trans\_from\_ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_intro}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{ax\_omega\_elements\_trans}} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop (Trans omega) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-048

Identifier	`thm_ord_omega`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Ord (ω ) 0\(\href{symbols/ord-pred.html}{\mathrm{Ord}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(snd ((plan_ord_def ω )) ⟨((ax_omega_trans_from_ind ((ax_ind_intro ω ) ⟨(snd ((plan_omega_def ∅)) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). fst (((ax_ind_elim Q ) hIndQ ))))), (λ (y :Class ). (λ (hyInOmega :(y ∈ ω )). (snd ((plan_omega_def SuccOrd (y ))) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). ((snd (((ax_ind_elim Q ) hIndQ )) y ) (((fst ((plan_omega_def y )) hyInOmega ) Q ) hIndQ )))))))⟩)) ax_omega_elements_trans ), ax_omega_elements_trans ⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_ord\_def}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_omega\_trans\_from\_ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_intro}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{ax\_omega\_elements\_trans}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{ax\_omega\_elements\_trans}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop (Ord omega) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-049

Identifier	`thm_omega_in_on`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0(ω ∈ On ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/in-pred.html}{\in} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0((ax_on_intro ω ) ⟨plan_omega_set , (snd ((plan_ord_def ω )) ⟨((ax_omega_trans_from_ind ((ax_ind_intro ω ) ⟨(snd ((plan_omega_def ∅)) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). fst (((ax_ind_elim Q ) hIndQ ))))), (λ (y :Class ). (λ (hyInOmega :(y ∈ ω )). (snd ((plan_omega_def SuccOrd (y ))) (λ (Q :Class ). (λ (hIndQ :Ind (Q )). ((snd (((ax_ind_elim Q ) hIndQ )) y ) (((fst ((plan_omega_def y )) hyInOmega ) Q ) hIndQ )))))))⟩)) ax_omega_elements_trans ), ax_omega_elements_trans ⟩)⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_on\_intro}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/variables.html}{\mathsf{plan\_omega\_set}} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_ord\_def}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_omega\_trans\_from\_ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_intro}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/typing-colon.html}{:} \href{symbols/ind-pred.html}{\mathrm{Ind}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ind\_elim}} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_omega\_def}} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hyInOmega}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Q} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIndQ}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{ax\_omega\_elements\_trans}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{ax\_omega\_elements\_trans}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop ((In omega) On) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-050

Identifier	`thm_on_ind`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0((∅ ∈ On ) ∧ (∀(a :Class ), ((a ∈ On ) → (SuccOrd (a ) ∈ On )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/in-pred.html}{\in} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/in-pred.html}{\in} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0⟨((ax_on_intro ∅) ⟨ax_empty_set , ax_empty_ord ⟩), (λ (a :Class ). (λ (hIn :(a ∈ On )). ((ax_on_intro SuccOrd (a )) ⟨((ax_ord_succ_set a ) fst (((ax_on_elim a ) hIn ))), ((ax_ord_succ_ord a ) ((ax_on_elim a ) hIn ))⟩)))⟩ 0\(\href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_on\_intro}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/variables.html}{\mathsf{ax\_empty\_set}} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{ax\_empty\_ord}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/in-pred.html}{\in} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_on\_intro}} \href{symbols/binunion-op.html}{\mathsf{SuccOrd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ord\_succ\_set}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_on\_elim}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_ord\_succ\_ord}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_on\_elim}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \)
SML Kernel AST	`(Sigma (_ : (Id Prop ((In EmptySet) On) true)) * (Pi (a : Class) -> (Pi (_ : (Id Prop ((In a) On) true)) -> (Id Prop ((In (SuccOrd a)) On) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-051

Identifier	`def_isCard`
Classification	Conservative Definition
Statement (Type)	0∀(C :Class ), ((Card (C ) → ∃(K :Class ), (CardOf (K ) = C )) ∧ ((∃(K :Class ), (CardOf (K ) = C )) → Card (C ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/card-op.html}{\mathsf{CardOf}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/card-op.html}{\mathsf{CardOf}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{K} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_card_def 0\(\href{symbols/variables.html}{\mathsf{plan\_card\_def}} \)
SML Kernel AST	`(Pi (C : Class) -> (Sigma (_ : (Pi (_ : (Card C)) -> (Sigma (K : Class) * (Id Class (CardOf K) C)))) * (Pi (_ : (Sigma (K : Class) * (Id Class (CardOf K) C))) -> (Card C))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-052

Identifier	`thm_omega_card`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Card (ω ) 0\(\href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(snd ((plan_card_def ω )) ⟨ω , ax_cardof_omega ⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_card\_def}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/omega-const.html}{\omega} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{ax\_cardof\_omega}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Card omega)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-053

Identifier	`thm_on_card`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0Card (On ) 0\(\href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(snd ((plan_card_def On )) ⟨On , ax_cardof_on ⟩) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_card\_def}} \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/on-const.html}{\mathbf{On}} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{ax\_cardof\_on}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Card On)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-054

Identifier	`thm_v_not_card`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0¬Card (V ) 0\(\href{symbols/neg-conn.html}{\neg} \href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (hCardV :Card (V )). ((ax_cardof_never_v fst ((fst ((plan_card_def V )) hCardV ))) snd ((fst ((plan_card_def V )) hCardV )))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hCardV}} \href{symbols/typing-colon.html}{:} \href{symbols/card-pred.html}{\mathrm{Card}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_cardof\_never\_v}} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_card\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hCardV}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_card\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hCardV}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (_ : (Card V)) -> False)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-055

Identifier	`ax_tarski_universes`
Classification	Irreducible MK/NBG Axiom
Statement (Type)	0∀(X :Class ), (set (X ) → ∃(U :Class ), (universe (U ) ∧ ((X ∈ U ) ∧ set (U )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{U} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universe-pred.html}{\mathsf{universe}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{U} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{U} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{U} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_tarski_axm 0\(\href{symbols/variables.html}{\mathsf{plan\_tarski\_axm}} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (_ : (Id Prop (Set X) true)) -> (Sigma (U : Class) * (Sigma (_ : (Universe U)) * (Sigma (_ : (Id Prop ((In X) U) true)) * (Id Prop (Set U) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-056

Identifier	`def_binary_union`
Classification	Conservative Definition
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ∀(z :Class ), (((z ∈ (A ∪ B )) → ∀(C :Prop ), (((z ∈ A ) → C ) → (((z ∈ B ) → C ) → C ))) ∧ ((∀(C :Prop ), (((z ∈ A ) → C ) → (((z ∈ B ) → C ) → C ))) → (z ∈ (A ∪ B )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_bin_union 0\(\href{symbols/variables.html}{\mathsf{plan\_bin\_union}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) ((BinUnion A) B)) true)) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : (Id Prop ((In z) A) true)) -> C)) -> (Pi (_ : (Pi (_ : (Id Prop ((In z) B) true)) -> C)) -> C))))) * (Pi (_ : (Pi (C : Prop) -> (Pi (_ : (Pi (_ : (Id Prop ((In z) A) true)) -> C)) -> (Pi (_ : (Pi (_ : (Id Prop ((In z) B) true)) -> C)) -> C)))) -> (Id Prop ((In z) ((BinUnion A) B)) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-057

Identifier	`def_binary_inter`
Classification	Conservative Definition
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ∀(z :Class ), (((z ∈ (A ∩ B )) → ((z ∈ A ) ∧ (z ∈ B ))) ∧ (((z ∈ A ) ∧ (z ∈ B )) → (z ∈ (A ∩ B )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_bin_inter 0\(\href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) ((BinInter A) B)) true)) -> (Sigma (_ : (Id Prop ((In z) A) true)) * (Id Prop ((In z) B) true)))) * (Pi (_ : (Sigma (_ : (Id Prop ((In z) A) true)) * (Id Prop ((In z) B) true))) -> (Id Prop ((In z) ((BinInter A) B)) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-058

Identifier	`def_unary_union`
Classification	Conservative Definition
Statement (Type)	0∀(A :Class ), ∀(z :Class ), (((z ∈ ⋃(A )) → ∃(y :Class ), ((y ∈ A ) ∧ (z ∈ y ))) ∧ ((∃(y :Class ), ((y ∈ A ) ∧ (z ∈ y ))) → (z ∈ ⋃(A )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/bigunion-op.html}{\bigcup} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/bigunion-op.html}{\bigcup} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_unary_union 0\(\href{symbols/variables.html}{\mathsf{plan\_unary\_union}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) (Union A)) true)) -> (Sigma (y : Class) * (Sigma (_ : (Id Prop ((In y) A) true)) * (Id Prop ((In z) y) true))))) * (Pi (_ : (Sigma (y : Class) * (Sigma (_ : (Id Prop ((In y) A) true)) * (Id Prop ((In z) y) true)))) -> (Id Prop ((In z) (Union A)) true)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-059

Identifier	`def_unary_inter`
Classification	Conservative Definition
Statement (Type)	0∀(A :Class ), ∀(z :Class ), (((z ∈ ⋂(A )) → ∀(y :Class ), ((y ∈ A ) → (z ∈ y ))) ∧ ((∀(y :Class ), ((y ∈ A ) → (z ∈ y ))) → (z ∈ ⋂(A )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/biginter-op.html}{\bigcap} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/biginter-op.html}{\bigcap} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_unary_inter 0\(\href{symbols/variables.html}{\mathsf{plan\_unary\_inter}} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) (Inter A)) true)) -> (Pi (y : Class) -> (Pi (_ : (Id Prop ((In y) A) true)) -> (Id Prop ((In z) y) true))))) * (Pi (_ : (Pi (y : Class) -> (Pi (_ : (Id Prop ((In y) A) true)) -> (Id Prop ((In z) y) true)))) -> (Id Prop ((In z) (Inter A)) true)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-060

Identifier	`def_singleton`
Classification	Conservative Definition
Statement (Type)	0∀(a :Class ), ∀(z :Class ), (((z ∈ {a }) → (set (a ) ∧ (z = a ))) ∧ ((set (a ) ∧ (z = a )) → (z ∈ {a }))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_singleton 0\(\href{symbols/variables.html}{\mathsf{plan\_singleton}} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) (Singleton a)) true)) -> (Sigma (_ : (Id Prop (Set a) true)) * (Id Class z a)))) * (Pi (_ : (Sigma (_ : (Id Prop (Set a) true)) * (Id Class z a))) -> (Id Prop ((In z) (Singleton a)) true)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-061

Identifier	`def_upair`
Classification	Conservative Definition
Statement (Type)	0∀(a :Class ), ∀(b :Class ), ∀(z :Class ), (((z ∈ {a , b }) → ∀(C :Prop ), (((z = a ) → C ) → (((z = b ) → C ) → C ))) ∧ ((∀(C :Prop ), (((z = a ) → C ) → (((z = b ) → C ) → C ))) → (z ∈ {a , b }))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0plan_upair 0\(\href{symbols/variables.html}{\mathsf{plan\_upair}} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) ((UPair a) b)) true)) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : (Id Class z a)) -> C)) -> (Pi (_ : (Pi (_ : (Id Class z b)) -> C)) -> C))))) * (Pi (_ : (Pi (C : Prop) -> (Pi (_ : (Pi (_ : (Id Class z a)) -> C)) -> (Pi (_ : (Pi (_ : (Id Class z b)) -> C)) -> C)))) -> (Id Prop ((In z) ((UPair a) b)) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-062

Identifier	`thm_russell_class_not_set`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0¬(∃(R :Class ), (set (R ) ∧ (∀(x :Class ), (((x ∈ R ) → (set (x ) ∧ ¬(x ∈ x ))) ∧ ((set (x ) ∧ ¬(x ∈ x )) → (x ∈ R )))))) 0\(\href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{R} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (hExists :∃(R :Class ), (set (R ) ∧ (∀(x :Class ), (((x ∈ R ) → (set (x ) ∧ ¬(x ∈ x ))) ∧ ((set (x ) ∧ ¬(x ∈ x )) → (x ∈ R )))))). ((λ (hIn :(fst (hExists ) ∈ fst (hExists ))). (snd ((fst ((snd (snd (hExists )) fst (hExists ))) hIn )) hIn )) (snd ((snd (snd (hExists )) fst (hExists ))) ⟨fst (snd (hExists )), (λ (hIn :(fst (hExists ) ∈ fst (hExists ))). (snd ((fst ((snd (snd (hExists )) fst (hExists ))) hIn )) hIn ))⟩))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/typing-colon.html}{:} \href{symbols/exists-binder.html}{\exists} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{R} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{R} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hExists}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hIn}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (_ : (Sigma (R : Class) * (Sigma (_ : (Id Prop (Set R) true)) * (Pi (x : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In x) R) true)) -> (Sigma (_ : (Id Prop (Set x) true)) * (Pi (_ : (Id Prop ((In x) x) true)) -> False)))) * (Pi (_ : (Sigma (_ : (Id Prop (Set x) true)) * (Pi (_ : (Id Prop ((In x) x) true)) -> False))) -> (Id Prop ((In x) R) true))))))) -> False)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-063

Identifier	`thm_dunaev_tagged_pair_proj`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ((pr1 ((A , B )) = A ) ∧ (pr2 ((A , B )) = B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/pr1-op.html}{\mathrm{pr}_1} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/pr2-op.html}{\mathrm{pr}_2} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). ⟨((ax_dunaev_pr1 A ) B ), ((ax_dunaev_pr2 A ) B )⟩)) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_dunaev\_pr1}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_dunaev\_pr2}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Sigma (_ : (Id Class (DunaevPr1 ((DunaevPair A) B)) A)) * (Id Class (DunaevPr2 ((DunaevPair A) B)) B))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-064

Identifier	`thm_dunaev_pair_injectivity`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ∀(C :Class ), ∀(D :Class ), (((A , B ) = (C , D )) → ((A = C ) ∧ (B = D ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{D} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{D} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (λ (C :Class ). (λ (D :Class ). (λ (hEq :((A , B ) = (C , D ))). ⟨(((((ax_eq_trans A ) pr1 ((C , D ))) C ) (((((ax_eq_trans A ) pr1 ((A , B ))) pr1 ((C , D ))) (((ax_eq_sym pr1 ((A , B ))) A ) fst ((((λ (A :Class ). (λ (B :Class ). ⟨((ax_dunaev_pr1 A ) B ), ((ax_dunaev_pr2 A ) B )⟩)) A ) B )))) (((ax_eq_cong_pr1 (A , B )) (C , D )) hEq ))) fst ((((λ (A :Class ). (λ (B :Class ). ⟨((ax_dunaev_pr1 A ) B ), ((ax_dunaev_pr2 A ) B )⟩)) C ) D ))), (((((ax_eq_trans B ) pr2 ((C , D ))) D ) (((((ax_eq_trans B ) pr2 ((A , B ))) pr2 ((C , D ))) (((ax_eq_sym pr2 ((A , B ))) B ) snd ((((λ (A :Class ). (λ (B :Class ). ⟨((ax_dunaev_pr1 A ) B ), ((ax_dunaev_pr2 A ) B )⟩)) A ) B )))) (((ax_eq_cong_pr2 (A , B )) (C , D )) hEq ))) snd ((((λ (A :Class ). 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SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (C : Class) -> (Pi (D : Class) -> (Pi (_ : (Id Class ((DunaevPair A) B) ((DunaevPair C) D))) -> (Sigma (_ : (Id Class A C)) * (Id Class B D)))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

4. Extended Set-Theoretic Algebra, Families & Dependent Projections (NBG-065–NBG-100)

Derived theorems for unions, intersections, singletons, unordered pairs, Sigma/Pi projections, and Dunaev beta-laws.

Num NBG Formulas-as-Types Record (Vertical Field Table)

NBG-065

Identifier	`thm_empty_set_elim`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), ((∀(z :Class ), ¬(z ∈ X )) → (X = ∅)) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (λ (h :∀(z :Class ), ¬(z ∈ X )). (((plan_extensionality X ) ∅) (λ (z :Class ). ⟨(λ (hz :(z ∈ X )). ((ax_false_elim (z ∈ ∅)) ((h z ) hz ))), (λ (hz :(z ∈ ∅)). ((ax_false_elim (z ∈ X )) ((ax_empty_no_elements z ) hz )))⟩)))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/neg-conn.html}{\neg} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_extensionality}} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_false\_elim}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_false\_elim}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_empty\_no\_elements}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> False))) -> (Id Class X EmptySet)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-066

Identifier	`thm_every_set_in_v`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(x :Class ), (set (x ) → (x ∈ V )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (x : Class) -> (Pi (_ : (Id Prop (Set x) true)) -> (Id Prop ((In x) V) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-067

Identifier	`thm_in_v_is_set`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(x :Class ), ((x ∈ V ) → set (x )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (x :Class ). fst (((fst ((plan_universal_class_def V )) refl (V )) x ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (x : Class) -> (Pi (_ : (Id Prop ((In x) V) true)) -> (Id Prop (Set x) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-068

Identifier	`thm_emptyset_in_v`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0(∅ ∈ V ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(((λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) ∅) ax_empty_set ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{ax\_empty\_set}} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop ((In EmptySet) V) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-069

Identifier	`thm_omega_in_v`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0(ω ∈ V ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(((λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) ω ) plan_omega_set ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{plan\_omega\_set}} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop ((In omega) V) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-070

Identifier	`thm_power_omega_is_set`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0set (𝒫 (ω )) 0\(\href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/power-op.html}{\mathcal{P}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0((plan_exponent ω ) plan_omega_set ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_exponent}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{plan\_omega\_set}} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop (Set (Power omega)) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-071

Identifier	`thm_singleton_self_in`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), (set (a ) → (a ∈ {a })) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (hSetA :set (a )). (snd (((plan_singleton a ) a )) ⟨hSetA , refl (a )⟩))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSetA}} \href{symbols/typing-colon.html}{:} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_singleton}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/variables.html}{\mathsf{hSetA}} \href{symbols/comma-delim.html}{,} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (_ : (Id Prop (Set a) true)) -> (Id Prop ((In a) (Singleton a)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-072

Identifier	`thm_upair_left_in`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), ∀(b :Class ), (a ∈ {a , b }) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/in-pred.html}{\in} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (b :Class ). (snd ((((plan_upair a ) b ) a )) (λ (C :Prop ). (λ (fa :((a = a ) → C )). (λ (fb :((a = b ) → C )). (fa refl (a )))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_upair}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((In a) ((UPair a) b)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-073

Identifier	`thm_upair_right_in`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), ∀(b :Class ), (b ∈ {a , b }) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/in-pred.html}{\in} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (b :Class ). (snd ((((plan_upair a ) b ) b )) (λ (C :Prop ). (λ (fa :((b = a ) → C )). (λ (fb :((b = b ) → C )). (fb refl (b )))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_upair}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((In b) ((UPair a) b)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-074

Identifier	`thm_inter_subset_left`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ((A ∩ B ) ⊆ A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) A )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). fst ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) A) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-075

Identifier	`thm_inter_subset_right`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ((A ∩ B ) ⊆ B ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) B )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). snd ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) B) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-076

Identifier	`thm_union_subset_left`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (A ⊆ (A ∪ B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def A ) (A ∪ B ))) (λ (z :Class ). (λ (hz :(z ∈ A )). (snd ((((plan_bin_union A ) B ) z )) (λ (C :Prop ). (λ (fa :((z ∈ A ) → C )). (λ (fb :((z ∈ B ) → C )). (fa hz )))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_union}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset A) ((BinUnion A) B)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-077

Identifier	`thm_union_subset_right`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (B ⊆ (A ∪ B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def B ) (A ∪ B ))) (λ (z :Class ). (λ (hz :(z ∈ B )). (snd ((((plan_bin_union A ) B ) z )) (λ (C :Prop ). (λ (fa :((z ∈ A ) → C )). (λ (fb :((z ∈ B ) → C )). (fb hz )))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_union}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset B) ((BinUnion A) B)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-078

Identifier	`thm_inter_idem_subset`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ((A ∩ A ) ⊆ A ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) A )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). fst ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) A )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Id Prop ((Subset ((BinInter A) A)) A) true))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-079

Identifier	`thm_union_idem_subset`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), (A ⊆ (A ∪ A )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def A ) (A ∪ B ))) (λ (z :Class ). (λ (hz :(z ∈ A )). (snd ((((plan_bin_union A ) B ) z )) (λ (C :Prop ). (λ (fa :((z ∈ A ) → C )). (λ (fb :((z ∈ B ) → C )). (fa hz )))))))))) A ) A )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_union}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Id Prop ((Subset A) ((BinUnion A) A)) true))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-080

Identifier	`thm_inter_set_left`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (set (A ) → set ((A ∩ B ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (λ (hSetA :set (A )). (((plan_heredity (A ∩ B )) A ) ⟨(((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) A )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). fst ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) B ), hSetA ⟩)))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSetA}} \href{symbols/typing-colon.html}{:} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_heredity}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{hSetA}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (_ : (Id Prop (Set A) true)) -> (Id Prop (Set ((BinInter A) B)) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-081

Identifier	`thm_semi_singleton_sub_char`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(s :Class ), ∀(A :Class ), ∀(x :Class ), ∀(y :Class ), (((s ⊆ A ) ∧ (∀(u :Class ), ∀(v :Class ), (((u ∈ s ) ∧ (v ∈ s )) → (u = v )))) → (((x ∈ s ) ∧ (y ∈ s )) → ((x ∈ A ) ∧ (x = y )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{v} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{v} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{v} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (s :Class ). (λ (A :Class ). (λ (x :Class ). (λ (y :Class ). (λ (hSSS :((s ⊆ A ) ∧ (∀(u :Class ), ∀(v :Class ), (((u ∈ s ) ∧ (v ∈ s )) → (u = v ))))). (λ (hXY :((x ∈ s ) ∧ (y ∈ s ))). ⟨(((fst (((plan_subclass_def s ) A )) fst (hSSS )) x ) fst (hXY )), (((snd (hSSS ) x ) y ) hXY )⟩)))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSSS}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{v} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{v} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{v} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hXY}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSSS}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hXY}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSSS}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hXY}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (s : Class) -> (Pi (A : Class) -> (Pi (x : Class) -> (Pi (y : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((Subset s) A) true)) * (Pi (u : Class) -> (Pi (v : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((In u) s) true)) * (Id Prop ((In v) s) true))) -> (Id Class u v)))))) -> (Pi (_ : (Sigma (_ : (Id Prop ((In x) s) true)) * (Id Prop ((In y) s) true))) -> (Sigma (_ : (Id Prop ((In x) A) true)) * (Id Class x y))))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-082

Identifier	`thm_family_value_eval`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(F :Class ), ∀(x :Class ), ∀(z :Class ), ∀(s :Class ), ∀(u :Class ), (((s ⊆ z ) ∧ ((u ∈ ((ValOf F ) x )) → (u ∈ s ))) → ((u ∈ ((ValOf F ) x )) → (u ∈ z ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (F :Class ). (λ (x :Class ). (λ (z :Class ). (λ (s :Class ). (λ (u :Class ). (λ (hFam :((s ⊆ z ) ∧ ((u ∈ ((ValOf F ) x )) → (u ∈ s )))). (λ (hUinVal :(u ∈ ((ValOf F ) x ))). (((fst (((plan_subclass_def s ) z )) fst (hFam )) u ) (snd (hFam ) hUinVal ))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hFam}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{s} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hUinVal}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{s} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hFam}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{u} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hFam}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hUinVal}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (F : Class) -> (Pi (x : Class) -> (Pi (z : Class) -> (Pi (s : Class) -> (Pi (u : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((Subset s) z) true)) * (Pi (_ : (Id Prop ((In u) ((ValOf F) x)) true)) -> (Id Prop ((In u) s) true)))) -> (Pi (_ : (Id Prop ((In u) ((ValOf F) x)) true)) -> (Id Prop ((In u) z) true))))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-083

Identifier	`thm_sigma_model_proj`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(B :Class ), ∀(x :Class ), ∀(y :Class ), ((((((OPair x ) y ) ∈ (Sigma B )) → ((x ∈ (Dom B )) ∧ (y ∈ ((ValOf B ) x )))) ∧ (((OPair x ) y ) ∈ (Sigma B ))) → ((x ∈ (Dom B )) ∧ (y ∈ ((ValOf B ) x )))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{OPair}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Sigma}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{OPair}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Sigma}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (B :Class ). (λ (x :Class ). (λ (y :Class ). (λ (h :(((((OPair x ) y ) ∈ (Sigma B )) → ((x ∈ (Dom B )) ∧ (y ∈ ((ValOf B ) x )))) ∧ (((OPair x ) y ) ∈ (Sigma B )))). (fst (h ) snd (h )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{OPair}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Sigma}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{OPair}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Sigma}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (B : Class) -> (Pi (x : Class) -> (Pi (y : Class) -> (Pi (_ : (Sigma (_ : (Pi (_ : (Id Prop ((In ((OPair x) y)) (Sigma B)) true)) -> (Sigma (_ : (Id Prop ((In x) (Dom B)) true)) * (Id Prop ((In y) ((ValOf B) x)) true)))) * (Id Prop ((In ((OPair x) y)) (Sigma B)) true))) -> (Sigma (_ : (Id Prop ((In x) (Dom B)) true)) * (Id Prop ((In y) ((ValOf B) x)) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-084

Identifier	`thm_pi_model_elim`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(B :Class ), ∀(f :Class ), ∀(x :Class ), (((∀(u :Class ), ((u ∈ (Dom B )) → (((ValOf f ) u ) ∈ ((ValOf B ) u )))) ∧ (x ∈ (Dom B ))) → (((ValOf f ) x ) ∈ ((ValOf B ) x ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{f} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{u} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{u} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{f} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (B :Class ). (λ (f :Class ). (λ (x :Class ). (λ (hPi :((∀(u :Class ), ((u ∈ (Dom B )) → (((ValOf f ) u ) ∈ ((ValOf B ) u )))) ∧ (x ∈ (Dom B )))). ((fst (hPi ) x ) snd (hPi )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{f} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hPi}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{u} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{f} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{u} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/valof-op.html}{\mathsf{ValOf}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{u} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/dom-op.html}{\mathrm{Dom}} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hPi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hPi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (B : Class) -> (Pi (f : Class) -> (Pi (x : Class) -> (Pi (_ : (Sigma (_ : (Pi (u : Class) -> (Pi (_ : (Id Prop ((In u) (Dom B)) true)) -> (Id Prop ((In ((ValOf f) u)) ((ValOf B) u)) true)))) * (Id Prop ((In x) (Dom B)) true))) -> (Id Prop ((In ((ValOf f) x)) ((ValOf B) x)) true)))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-085

Identifier	`thm_if_dep_true_branch`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(Phi :Prop ), ∀(T1 :(Phi → Prop )), ∀(T2 :(¬Phi → Prop )), (((∀(h :Phi ), (T1 h )) ∧ (∀(nh :¬Phi ), (T2 nh ))) → ∀(h :Phi ), (T1 h )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (Phi :Prop ). (λ (T1 :(Phi → Prop )). (λ (T2 :(¬Phi → Prop )). (λ (branches :((∀(h :Phi ), (T1 h )) ∧ (∀(nh :¬Phi ), (T2 nh )))). (λ (h :Phi ). (fst (branches ) h )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{branches}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{branches}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (Phi : Prop) -> (Pi (T1 : (Pi (_ : Phi) -> Prop)) -> (Pi (T2 : (Pi (_ : (Pi (_ : Phi) -> False)) -> Prop)) -> (Pi (_ : (Sigma (_ : (Pi (h : Phi) -> (T1 h))) * (Pi (nh : (Pi (_ : Phi) -> False)) -> (T2 nh)))) -> (Pi (h : Phi) -> (T1 h))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-086

Identifier	`thm_if_dep_false_branch`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(Phi :Prop ), ∀(T1 :(Phi → Prop )), ∀(T2 :(¬Phi → Prop )), (((∀(h :Phi ), (T1 h )) ∧ (∀(nh :¬Phi ), (T2 nh ))) → ∀(nh :¬Phi ), (T2 nh )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (Phi :Prop ). (λ (T1 :(Phi → Prop )). (λ (T2 :(¬Phi → Prop )). (λ (branches :((∀(h :Phi ), (T1 h )) ∧ (∀(nh :¬Phi ), (T2 nh )))). (λ (nh :¬Phi ). (snd (branches ) nh )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{branches}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T1}} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{T2}} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/typing-colon.html}{:} \href{symbols/neg-conn.html}{\neg} \href{symbols/variables.html}{\mathsf{Phi}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{branches}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{nh}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (Phi : Prop) -> (Pi (T1 : (Pi (_ : Phi) -> Prop)) -> (Pi (T2 : (Pi (_ : (Pi (_ : Phi) -> False)) -> Prop)) -> (Pi (_ : (Sigma (_ : (Pi (h : Phi) -> (T1 h))) * (Pi (nh : (Pi (_ : Phi) -> False)) -> (T2 nh)))) -> (Pi (nh : (Pi (_ : Phi) -> False)) -> (T2 nh))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-087

Identifier	`thm_model_stratification_trans`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(W :Class ), ∀(E :Class ), ∀(T :Class ), ∀(F :Class ), ((((mu_A W ) ⊆ (mu_A E )) ∧ (((mu_A E ) ⊆ (mu_A T )) ∧ ((mu_A T ) ⊆ (mu_A F )))) → ((mu_A W ) ⊆ (mu_A F ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{W} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{E} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{T} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (W :Class ). (λ (E :Class ). (λ (T :Class ). (λ (F :Class ). (λ (hStrat :(((mu_A W ) ⊆ (mu_A E )) ∧ (((mu_A E ) ⊆ (mu_A T )) ∧ ((mu_A T ) ⊆ (mu_A F ))))). (snd (((plan_subclass_def (mu_A W )) (mu_A F ))) (λ (z :Class ). (λ (hz :(z ∈ (mu_A W ))). (((fst (((plan_subclass_def (mu_A T )) (mu_A F ))) snd (snd (hStrat ))) z ) (((fst (((plan_subclass_def (mu_A E )) (mu_A T ))) fst (snd (hStrat ))) z ) (((fst (((plan_subclass_def (mu_A W )) (mu_A E ))) fst (hStrat )) z ) hz ))))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{W} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{E} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{T} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{F} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hStrat}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{F} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hStrat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{T} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hStrat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{W} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/mu-interpretation.html}{\mu_A} \href{symbols/variables.html}{E} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hStrat}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (W : Class) -> (Pi (E : Class) -> (Pi (T : Class) -> (Pi (F : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((Subset (mu_A W)) (mu_A E)) true)) * (Sigma (_ : (Id Prop ((Subset (mu_A E)) (mu_A T)) true)) * (Id Prop ((Subset (mu_A T)) (mu_A F)) true)))) -> (Id Prop ((Subset (mu_A W)) (mu_A F)) true))))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-088

Identifier	`thm_dunaev_pair_fst_beta`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(x :Class ), ∀(y :Class ), (pr1 ((x , y )) = x ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/pr1-op.html}{\mathrm{pr}_1} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (x :Class ). (λ (y :Class ). ((ax_dunaev_pr1 x ) y ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_dunaev\_pr1}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (x : Class) -> (Pi (y : Class) -> (Id Class (DunaevPr1 ((DunaevPair x) y)) x)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-089

Identifier	`thm_dunaev_pair_snd_beta`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(x :Class ), ∀(y :Class ), (pr2 ((x , y )) = y ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/pr2-op.html}{\mathrm{pr}_2} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (x :Class ). (λ (y :Class ). ((ax_dunaev_pr2 x ) y ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_dunaev\_pr2}} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (x : Class) -> (Pi (y : Class) -> (Id Class (DunaevPr2 ((DunaevPair x) y)) y)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-090

Identifier	`thm_singleton_elem_eq`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), ∀(z :Class ), ((z ∈ {a }) → (z = a )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (z :Class ). (λ (hz :(z ∈ {a })). snd ((fst (((plan_singleton a ) z )) hz ))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_singleton}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) (Singleton a)) true)) -> (Id Class z a))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-091

Identifier	`thm_singleton_subset_upair_left`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), ∀(b :Class ), ({a } ⊆ {a , b }) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (b :Class ). (snd (((plan_subclass_def {a }) {a , b })) (λ (z :Class ). (λ (hz :(z ∈ {a })). (snd ((((plan_upair a ) b ) z )) (λ (C :Prop ). (λ (fa :((z = a ) → C )). (λ (fb :((z = b ) → C )). (fa ((((λ (a :Class ). (λ (z :Class ). (λ (hz :(z ∈ {a })). snd ((fst (((plan_singleton a ) z )) hz ))))) a ) z ) hz ))))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_upair}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_singleton}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((Subset (Singleton a)) ((UPair a) b)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-092

Identifier	`thm_singleton_subset_upair_right`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), ∀(b :Class ), ({b } ⊆ {a , b }) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{b} \href{symbols/singleton-op.html}{\}} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (b :Class ). (snd (((plan_subclass_def {b }) {a , b })) (λ (z :Class ). (λ (hz :(z ∈ {b })). (snd ((((plan_upair a ) b ) z )) (λ (C :Prop ). (λ (fa :((z = a ) → C )). (λ (fb :((z = b ) → C )). (fb ((((λ (a :Class ). (λ (z :Class ). (λ (hz :(z ∈ {a })). snd ((fst (((plan_singleton a ) z )) hz ))))) b ) z ) hz ))))))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{b} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{b} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/upair-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{b} \href{symbols/upair-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{b} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_upair}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{C} \href{symbols/typing-colon.html}{:} \href{symbols/prop-bool.html}{\mathsf{Prop}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fb}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_singleton}} \href{symbols/variables.html}{a} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{b} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((Subset (Singleton b)) ((UPair a) b)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-093

Identifier	`thm_inter_subset_union`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), ((A ∩ B ) ⊆ (A ∪ B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) (A ∪ B ))) ((((((λ (X :Class ). (λ (Y :Class ). (λ (Z :Class ). (λ (h1 :∀(z :Class ), ((z ∈ X ) → (z ∈ Y ))). (λ (h2 :∀(z :Class ), ((z ∈ Y ) → (z ∈ Z ))). (λ (z :Class ). (λ (hz :(z ∈ X )). ((h2 z ) ((h1 z ) hz ))))))))) (A ∩ B )) A ) (A ∪ B )) (fst (((plan_subclass_def (A ∩ B )) A )) (((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) A )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). fst ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) B ))) (fst (((plan_subclass_def A ) (A ∪ B ))) (((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def A ) (A ∪ B ))) (λ (z :Class ). (λ (hz :(z ∈ A )). (snd ((((plan_bin_union A ) B ) z )) (λ (C :Prop ). (λ (fa :((z ∈ A ) → C )). (λ (fb :((z ∈ B ) → C )). (fa hz )))))))))) A ) B )))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/binunion-op.html}{\cup} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{h1}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} 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\href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/variables.html}{C} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{fa}} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) ((BinUnion A) B)) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-094

Identifier	`thm_inter_set_right`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (set (B ) → set ((A ∩ B ))) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). (λ (hSetB :set (B )). (((plan_heredity (A ∩ B )) B ) ⟨(((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) B )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). snd ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) B ), hSetB ⟩)))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSetB}} \href{symbols/typing-colon.html}{:} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_heredity}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/variables.html}{\mathsf{hSetB}} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Pi (_ : (Id Prop (Set B) true)) -> (Id Prop (Set ((BinInter A) B)) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-095

Identifier	`thm_emptyset_subset_all`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), (∅ ⊆ X ) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (snd (((plan_subclass_def ∅) X )) ((λ (X :Class ). (λ (z :Class ). (λ (h :(z ∈ ∅)). ((ax_false_elim (z ∈ X )) ((ax_empty_no_elements z ) h ))))) X ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{h} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/emptyset-const.html}{\emptyset} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_false\_elim}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{ax\_empty\_no\_elements}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{h} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Id Prop ((Subset EmptySet) X) true))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-096

Identifier	`thm_subset_antisym`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(X :Class ), ∀(Y :Class ), (((X ⊆ Y ) ∧ (Y ⊆ X )) → (X = Y )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/eq-pred.html}{=} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (X :Class ). (λ (Y :Class ). (λ (hSubPair :((X ⊆ Y ) ∧ (Y ⊆ X ))). (((plan_extensionality X ) Y ) (λ (z :Class ). ⟨((fst (((plan_subclass_def X ) Y )) fst (hSubPair )) z ), ((fst (((plan_subclass_def Y ) X )) snd (hSubPair )) z )⟩))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSubPair}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{X} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{Y} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_extensionality}} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSubPair}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{Y} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{X} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSubPair}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (X : Class) -> (Pi (Y : Class) -> (Pi (_ : (Sigma (_ : (Id Prop ((Subset X) Y) true)) * (Id Prop ((Subset Y) X) true))) -> (Id Class X Y))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-097

Identifier	`thm_every_set_subset_v`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ((∀(z :Class ), ((z ∈ A ) → set (z ))) → (A ⊆ V )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (hElemSet :∀(z :Class ), ((z ∈ A ) → set (z ))). (snd (((plan_subclass_def A ) V )) (λ (z :Class ). (λ (hz :(z ∈ A )). (((λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) z ) ((hElemSet z ) hz ))))))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hElemSet}} \href{symbols/typing-colon.html}{:} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hElemSet}} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) A) true)) -> (Id Prop (Set z) true)))) -> (Id Prop ((Subset A) V) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-098

Identifier	`thm_singleton_in_v_if_set`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(a :Class ), (set ({a }) → ({a } ∈ V )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/impl-conn.html}{\rightarrow} \href{symbols/delimiters-brackets.html}{(} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (a :Class ). (λ (hSetSing :set ({a })). (((λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) {a }) hSetSing ))) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{a} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hSetSing}} \href{symbols/typing-colon.html}{:} \href{symbols/set-pred.html}{\mathsf{set}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/singleton-op.html}{\{} \href{symbols/variables.html}{a} \href{symbols/singleton-op.html}{\}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hSetSing}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (a : Class) -> (Pi (_ : (Id Prop (Set (Singleton a)) true)) -> (Id Prop ((In (Singleton a)) V) true)))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-099

Identifier	`thm_power_omega_in_v`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0(𝒫 (ω ) ∈ V ) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/power-op.html}{\mathcal{P}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/in-pred.html}{\in} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(((λ (x :Class ). snd (((fst ((plan_universal_class_def V )) refl (V )) x ))) 𝒫 (ω )) ((plan_exponent ω ) plan_omega_set )) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{x} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_universal\_class\_def}} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/eq-pred.html}{\mathsf{refl}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/universal-class-const.html}{\mathbf{V}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{x} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/power-op.html}{\mathcal{P}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_exponent}} \href{symbols/omega-const.html}{\omega} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{plan\_omega\_set}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Id Prop ((In (Power omega)) V) true)`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)

NBG-100

Identifier	`thm_inter_subset_both`
Classification	Derived Theorem (Constructive λ-Proof)
Statement (Type)	0∀(A :Class ), ∀(B :Class ), (((A ∩ B ) ⊆ A ) ∧ ((A ∩ B ) ⊆ B )) 0\(\href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/forall-binder.html}{\forall} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/and-conn.html}{\land} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/sub-pred.html}{\subseteq} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
Proof Program	0(λ (A :Class ). (λ (B :Class ). ⟨(((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) A )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). fst ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) B ), (((λ (A :Class ). (λ (B :Class ). (snd (((plan_subclass_def (A ∩ B )) B )) (λ (z :Class ). (λ (hz :(z ∈ (A ∩ B ))). snd ((fst ((((plan_bin_inter A ) B ) z )) hz ))))))) A ) B )⟩)) 0\(\href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/antiquotation.html}{\langle } \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/comma-delim.html}{,} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{B} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_subclass\_def}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/typing-colon.html}{:} \href{symbols/class-type.html}{\mathsf{Class}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/delimiters-brackets.html}{(} \href{symbols/lambda-binder.html}{\lambda} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/typing-colon.html}{:} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{z} \href{symbols/in-pred.html}{\in} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{A} \href{symbols/bininter-op.html}{\cap} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/dot-delim.html}{.} \href{symbols/snd-op.html}{\mathsf{snd}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/fst-op.html}{\mathsf{fst}} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/delimiters-brackets.html}{(} \href{symbols/variables.html}{\mathsf{plan\_bin\_inter}} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{z} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{\mathsf{hz}} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{A} \href{symbols/delimiters-brackets.html}{)} \href{symbols/variables.html}{B} \href{symbols/delimiters-brackets.html}{)} \href{symbols/antiquotation.html}{\rangle } \href{symbols/delimiters-brackets.html}{)} \href{symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`(Pi (A : Class) -> (Pi (B : Class) -> (Sigma (_ : (Id Prop ((Subset ((BinInter A) B)) A) true)) * (Id Prop ((Subset ((BinInter A) B)) B) true))))`
Verification	✔ Mechanically Type-Checked (DependentTypeChecker.check)