{
  "_provenance": "AUTO-GENERATED by set_theory_prover/program/nbg_html_generator.ML from verified_theorems.ML. DO NOT EDIT BY AI OR HAND.",
  "generator": "set_theory_prover/program/nbg_html_generator.ML",
  "source": "set_theory_prover/program/verified_theorems.ML",
  "total_count": 100,
  "verified_count": 100,
  "derived_count": 77,
  "definitional_count": 16,
  "axiomatic_count": 7,
  "theorems": [
    {
      "id": 1,
      "num": "NBG-001",
      "name": "thm_A01",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), (A → (B → A))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(a:A). (λ(b:B). a))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (_ : B) -> A))))"
    },
    {
      "id": 2,
      "num": "NBG-002",
      "name": "thm_A02",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ∀(C:Prop), ((A → (B → C)) → ((A → B) → (A → C)))",
      "proof_unicode": "(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))))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> (Pi (_ : B) -> C))) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : A) -> C))))))"
    },
    {
      "id": 3,
      "num": "NBG-003",
      "name": "thm_A03",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A ∧ B) → A)",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(p:(A ∧ B)). fst(p)))) ≡ (λ(A:Prop). (λ(B:Prop). fst))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> A)))"
    },
    {
      "id": 4,
      "num": "NBG-004",
      "name": "thm_A04",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A ∧ B) → B)",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(p:(A ∧ B)). snd(p))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> B)))"
    },
    {
      "id": 5,
      "num": "NBG-005",
      "name": "thm_A05",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), (A → (B → (A ∧ B)))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(a:A). (λ(b:B). ⟨a, b⟩))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (_ : B) -> (Sigma (_ : A) * B)))))"
    },
    {
      "id": 6,
      "num": "NBG-006",
      "name": "thm_A06",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), (A → ∀(C:Prop), ((A → C) → ((B → C) → C)))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(a:A). (λ(C:Prop). (λ(fa:(A → C)). (λ(fb:(B → C)). (fa a)))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : A) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> C)) -> (Pi (_ : (Pi (_ : B) -> C)) -> C))))))"
    },
    {
      "id": 7,
      "num": "NBG-007",
      "name": "thm_A07",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), (B → ∀(C:Prop), ((A → C) → ((B → C) → C)))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(b:B). (λ(C:Prop). (λ(fa:(A → C)). (λ(fb:(B → C)). (fb b)))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : B) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> C)) -> (Pi (_ : (Pi (_ : B) -> C)) -> C))))))"
    },
    {
      "id": 8,
      "num": "NBG-008",
      "name": "thm_A08",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ∀(C:Prop), ((A → C) → ((B → C) → (∀(D:Prop), ((A → D) → ((B → D) → D)) → C)))",
      "proof_unicode": "(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)))))))",
      "statement_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))))))"
    },
    {
      "id": 9,
      "num": "NBG-009",
      "name": "thm_A09",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A → B) → ((A → ¬B) → ¬A))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(f:(A → B)). (λ(g:(A → ¬B)). (λ(a:A). ((g a) (f a)))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : A) -> (Pi (_ : B) -> False))) -> (Pi (_ : A) -> False)))))"
    },
    {
      "id": 10,
      "num": "NBG-010",
      "name": "thm_A10",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), (¬¬A → A)",
      "proof_unicode": "(λ(A:Prop). (λ(dneg:¬¬A). ((ax_dneg_elim A) dneg)))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (_ : (Pi (_ : (Pi (_ : A) -> False)) -> False)) -> A))"
    },
    {
      "id": 11,
      "num": "NBG-011",
      "name": "thm_A11",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), (⊥ → A)",
      "proof_unicode": "(λ(A:Prop). (λ(falsum:⊥). ((ax_false_elim A) falsum)))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (_ : False) -> A))"
    },
    {
      "id": 12,
      "num": "NBG-012",
      "name": "thm_A12",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), (((A → B) → A) → A)",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(h:((A → B) → A)). (((ax_peirce A) B) h))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : (Pi (_ : A) -> B)) -> A)) -> A)))"
    },
    {
      "id": 13,
      "num": "NBG-013",
      "name": "thm_identity",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), (A → A)",
      "proof_unicode": "(λ(A:Prop). (λ(x:A). x))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (_ : A) -> A))"
    },
    {
      "id": 14,
      "num": "NBG-014",
      "name": "thm_modus_ponens",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A → B) → (A → B))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(f:(A → B)). (λ(a:A). (f a)))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : A) -> B))))"
    },
    {
      "id": 15,
      "num": "NBG-015",
      "name": "thm_transitivity",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ∀(C:Prop), ((A → B) → ((B → C) → (A → C)))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(C:Prop). (λ(f:(A → B)). (λ(g:(B → C)). (λ(a:A). (g (f a))))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (C : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : B) -> C)) -> (Pi (_ : A) -> C))))))"
    },
    {
      "id": 16,
      "num": "NBG-016",
      "name": "thm_and_comm",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A ∧ B) → (B ∧ A))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(p:(A ∧ B)). ⟨snd(p), fst(p)⟩)))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Sigma (_ : A) * B)) -> (Sigma (_ : B) * A))))"
    },
    {
      "id": 17,
      "num": "NBG-017",
      "name": "thm_contraposition",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:Prop), ((A → B) → (¬B → ¬A))",
      "proof_unicode": "(λ(A:Prop). (λ(B:Prop). (λ(f:(A → B)). (λ(nb:¬B). (λ(a:A). (nb (f a)))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : Prop) -> (Pi (_ : (Pi (_ : A) -> B)) -> (Pi (_ : (Pi (_ : B) -> False)) -> (Pi (_ : A) -> False)))))"
    },
    {
      "id": 18,
      "num": "NBG-018",
      "name": "thm_D01",
      "kind": "Derived",
      "statement_unicode": "∀(A:Prop), ∀(B:(Class → Prop)), (∀(x:Class), (A → (B x)) → (A → ∀(x:Class), (B x)))",
      "proof_unicode": "(λ(A:Prop). (λ(B:(Class → Prop)). (λ(h:∀(x:Class), (A → (B x))). (λ(a:A). (λ(x:Class). ((h x) a))))))",
      "statement_ast": "(Pi (A : Prop) -> (Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (_ : (Pi (x : Class) -> (Pi (_ : A) -> (B x)))) -> (Pi (_ : A) -> (Pi (x : Class) -> (B x))))))"
    },
    {
      "id": 19,
      "num": "NBG-019",
      "name": "thm_forall_inst",
      "kind": "Derived",
      "statement_unicode": "∀(B:(Class → Prop)), ∀(c:Class), (∀(x:Class), (B x) → (B c))",
      "proof_unicode": "(λ(B:(Class → Prop)). (λ(c:Class). (λ(h:∀(x:Class), (B x)). (h c))))",
      "statement_ast": "(Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (c : Class) -> (Pi (_ : (Pi (x : Class) -> (B x))) -> (B c))))"
    },
    {
      "id": 20,
      "num": "NBG-020",
      "name": "thm_exists_intro",
      "kind": "Derived",
      "statement_unicode": "∀(B:(Class → Prop)), ∀(c:Class), ((B c) → ∃(x:Class), (B x))",
      "proof_unicode": "(λ(B:(Class → Prop)). (λ(c:Class). (λ(witness:(B c)). ⟨c, witness⟩)))",
      "statement_ast": "(Pi (B : (Pi (_ : Class) -> Prop)) -> (Pi (c : Class) -> (Pi (_ : (B c)) -> (Sigma (x : Class) * (B x)))))"
    },
    {
      "id": 21,
      "num": "NBG-021",
      "name": "def_isSet",
      "kind": "Definitional",
      "statement_unicode": "∀(X:Class), ((set(X) → ∃(Y:Class), (X ∈ Y)) ∧ (∃(Y:Class), (X ∈ Y) → set(X)))",
      "proof_unicode": "plan_isSet_def",
      "statement_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))))"
    },
    {
      "id": 22,
      "num": "NBG-022",
      "name": "def_isProper",
      "kind": "Definitional",
      "statement_unicode": "∀(K:Class), ((proper(K) → ¬set(K)) ∧ (¬set(K) → proper(K)))",
      "proof_unicode": "plan_isProper_def",
      "statement_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))))"
    },
    {
      "id": 23,
      "num": "NBG-023",
      "name": "def_empty_set",
      "kind": "Definitional",
      "statement_unicode": "∀(X:Class), (((X = ∅) → ∀(z:Class), ¬(z ∈ X)) ∧ (∀(z:Class), ¬(z ∈ X) → (X = ∅)))",
      "proof_unicode": "plan_empty_set_def",
      "statement_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))))"
    },
    {
      "id": 24,
      "num": "NBG-024",
      "name": "def_universal_class",
      "kind": "Definitional",
      "statement_unicode": "∀(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)))",
      "proof_unicode": "plan_universal_class_def",
      "statement_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))))"
    },
    {
      "id": 25,
      "num": "NBG-025",
      "name": "def_subclass",
      "kind": "Definitional",
      "statement_unicode": "∀(X:Class), ∀(Y:Class), (((X ⊆ Y) → ∀(z:Class), ((z ∈ X) → (z ∈ Y))) ∧ (∀(z:Class), ((z ∈ X) → (z ∈ Y)) → (X ⊆ Y)))",
      "proof_unicode": "plan_subclass_def",
      "statement_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)))))"
    },
    {
      "id": 26,
      "num": "NBG-026",
      "name": "thm_emptyset_is_subclass",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), ∀(z:Class), ((z ∈ ∅) → (z ∈ X))",
      "proof_unicode": "(λ(X:Class). (λ(z:Class). (λ(h:(z ∈ ∅)). ((ax_false_elim (z ∈ X)) ((ax_empty_no_elements z) h)))))",
      "statement_ast": "(Pi (X : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) EmptySet) true)) -> (Id Prop ((In z) X) true))))"
    },
    {
      "id": 27,
      "num": "NBG-027",
      "name": "ax_extensionality",
      "kind": "Axiomatic",
      "statement_unicode": "∀(X:Class), ∀(Y:Class), (∀(z:Class), (((z ∈ X) → (z ∈ Y)) ∧ ((z ∈ Y) → (z ∈ X))) → (X = Y))",
      "proof_unicode": "plan_extensionality",
      "statement_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))))"
    },
    {
      "id": 28,
      "num": "NBG-028",
      "name": "ax_heredity",
      "kind": "Axiomatic",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (((A ⊆ B) ∧ set(B)) → set(A))",
      "proof_unicode": "plan_heredity",
      "statement_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))))"
    },
    {
      "id": 29,
      "num": "NBG-029",
      "name": "ax_exponent",
      "kind": "Axiomatic",
      "statement_unicode": "∀(x:Class), (set(x) → set(𝒫(x)))",
      "proof_unicode": "plan_exponent",
      "statement_ast": "(Pi (x : Class) -> (Pi (_ : (Id Prop (Set x) true)) -> (Id Prop (Set (Power x)) true)))"
    },
    {
      "id": 30,
      "num": "NBG-030",
      "name": "ax_regularity",
      "kind": "Axiomatic",
      "statement_unicode": "∀(X:Class), ((𝒫(X) ⊆ X) → (V ⊆ X))",
      "proof_unicode": "plan_regularity",
      "statement_ast": "(Pi (X : Class) -> (Pi (_ : (Id Prop ((Subset (Power X)) X) true)) -> (Id Prop ((Subset V) X) true)))"
    },
    {
      "id": 31,
      "num": "NBG-031",
      "name": "def_ord",
      "kind": "Definitional",
      "statement_unicode": "∀(A:Class), ((Ord(A) → (Tr(A) ∧ ∀(B:Class), ((B ∈ A) → Tr(B)))) ∧ ((Tr(A) ∧ ∀(B:Class), ((B ∈ A) → Tr(B))) → Ord(A)))",
      "proof_unicode": "plan_ord_def",
      "statement_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))))"
    },
    {
      "id": 32,
      "num": "NBG-032",
      "name": "ax_global_choice",
      "kind": "Axiomatic",
      "statement_unicode": "∀(A:Class), (∃(x:Class), (x ∈ A) → (choice(A) ∈ A))",
      "proof_unicode": "plan_global_choice",
      "statement_ast": "(Pi (A : Class) -> (Pi (_ : (Sigma (x : Class) * (Id Prop ((In x) A) true))) -> (Id Prop ((In (choice A)) A) true)))"
    },
    {
      "id": 33,
      "num": "NBG-033",
      "name": "thm_subset_refl",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), ∀(z:Class), ((z ∈ X) → (z ∈ X))",
      "proof_unicode": "(λ(X:Class). (λ(z:Class). (λ(hz:(z ∈ X)). hz)))",
      "statement_ast": "(Pi (X : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> (Id Prop ((In z) X) true))))"
    },
    {
      "id": 34,
      "num": "NBG-034",
      "name": "thm_subset_trans",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), ∀(Y:Class), ∀(Z:Class), (∀(z:Class), ((z ∈ X) → (z ∈ Y)) → (∀(z:Class), ((z ∈ Y) → (z ∈ Z)) → ∀(z:Class), ((z ∈ X) → (z ∈ Z))))",
      "proof_unicode": "(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)))))))))",
      "statement_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))))))))"
    },
    {
      "id": 35,
      "num": "NBG-035",
      "name": "def_dunaev_opair",
      "kind": "Definitional",
      "statement_unicode": "∀(a:Class), ∀(b:Class), ((a, b) = (a, b))",
      "proof_unicode": "def_dunaev_opair_witness",
      "statement_ast": "(Pi (a : Class) -> (Pi (b : Class) -> (Id Class ((Pair a) b) ((DunaevPair a) b))))"
    },
    {
      "id": 36,
      "num": "NBG-036",
      "name": "def_relational_func",
      "kind": "Definitional",
      "statement_unicode": "∀(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)))",
      "proof_unicode": "plan_relational_func_def",
      "statement_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))))"
    },
    {
      "id": 37,
      "num": "NBG-037",
      "name": "thm_zero_is_nat",
      "kind": "Derived",
      "statement_unicode": "(0 = 0)",
      "proof_unicode": "refl(0)",
      "statement_ast": "(Id Nat 0 0)"
    },
    {
      "id": 38,
      "num": "NBG-038",
      "name": "thm_succ_preserves_nat",
      "kind": "Derived",
      "statement_unicode": "∀(n:Nat), (succ(n) = succ(n))",
      "proof_unicode": "(λ(n:Nat). refl(succ(n)))",
      "statement_ast": "(Pi (n : Nat) -> (Id Nat (succ n) (succ n)))"
    },
    {
      "id": 39,
      "num": "NBG-039",
      "name": "thm_zero_not_succ",
      "kind": "Derived",
      "statement_unicode": "∀(n:Nat), ∀(h:(succ(n) = 0)), ⊥",
      "proof_unicode": "(λ(n:Nat). (λ(h:(succ(n) = 0)). ((ax_peano_zero_not_succ n) h)))",
      "statement_ast": "(Pi (n : Nat) -> (Pi (h : (Id Nat (succ n) 0)) -> False))"
    },
    {
      "id": 40,
      "num": "NBG-040",
      "name": "thm_succ_injective",
      "kind": "Derived",
      "statement_unicode": "∀(n:Nat), ∀(m:Nat), ∀(h:(succ(n) = succ(m))), (n = m)",
      "proof_unicode": "(λ(n:Nat). (λ(m:Nat). (λ(h:(succ(n) = succ(m))). (((ax_peano_succ_inj n) m) h))))",
      "statement_ast": "(Pi (n : Nat) -> (Pi (m : Nat) -> (Pi (h : (Id Nat (succ n) (succ m))) -> (Id Nat n m))))"
    },
    {
      "id": 41,
      "num": "NBG-041",
      "name": "thm_add_zero_left",
      "kind": "Derived",
      "statement_unicode": "∀(n:Nat), (n = n)",
      "proof_unicode": "(λ(n:Nat). refl(n))",
      "statement_ast": "(Pi (n : Nat) -> (Id Nat n n))"
    },
    {
      "id": 42,
      "num": "NBG-042",
      "name": "thm_mathematical_induction",
      "kind": "Derived",
      "statement_unicode": "∀(P:(Nat → Prop)), ∀(base:(P 0)), ∀(step:∀(n:Nat), ∀(ih:(P n)), (P succ(n))), ∀(n:Nat), (P n)",
      "proof_unicode": "(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)))))",
      "statement_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)))))"
    },
    {
      "id": 43,
      "num": "NBG-043",
      "name": "thm_v_ind",
      "kind": "Derived",
      "statement_unicode": "Ind(V)",
      "proof_unicode": "((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)))))⟩)",
      "statement_ast": "(Id Prop (Ind V) true)"
    },
    {
      "id": 44,
      "num": "NBG-044",
      "name": "thm_omega_def",
      "kind": "Definitional",
      "statement_unicode": "∀(x:Class), (((x ∈ ω) → ∀(Q:Class), (Ind(Q) → (x ∈ Q))) ∧ (∀(Q:Class), (Ind(Q) → (x ∈ Q)) → (x ∈ ω)))",
      "proof_unicode": "plan_omega_def",
      "statement_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))))"
    },
    {
      "id": 45,
      "num": "NBG-045",
      "name": "thm_omega_set",
      "kind": "Axiomatic",
      "statement_unicode": "set(ω)",
      "proof_unicode": "plan_omega_set",
      "statement_ast": "(Id Prop (Set omega) true)"
    },
    {
      "id": 46,
      "num": "NBG-046",
      "name": "thm_omega_ind",
      "kind": "Derived",
      "statement_unicode": "Ind(ω)",
      "proof_unicode": "((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)))))))⟩)",
      "statement_ast": "(Id Prop (Ind omega) true)"
    },
    {
      "id": 47,
      "num": "NBG-047",
      "name": "thm_tr_omega",
      "kind": "Derived",
      "statement_unicode": "Tr(ω)",
      "proof_unicode": "((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)",
      "statement_ast": "(Id Prop (Trans omega) true)"
    },
    {
      "id": 48,
      "num": "NBG-048",
      "name": "thm_ord_omega",
      "kind": "Derived",
      "statement_unicode": "Ord(ω)",
      "proof_unicode": "(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⟩)",
      "statement_ast": "(Id Prop (Ord omega) true)"
    },
    {
      "id": 49,
      "num": "NBG-049",
      "name": "thm_omega_in_on",
      "kind": "Derived",
      "statement_unicode": "(ω ∈ On)",
      "proof_unicode": "((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⟩)⟩)",
      "statement_ast": "(Id Prop ((In omega) On) true)"
    },
    {
      "id": 50,
      "num": "NBG-050",
      "name": "thm_on_ind",
      "kind": "Derived",
      "statement_unicode": "((∅ ∈ On) ∧ ∀(a:Class), ((a ∈ On) → (SuccOrd(a) ∈ On)))",
      "proof_unicode": "⟨((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))⟩)))⟩",
      "statement_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))))"
    },
    {
      "id": 51,
      "num": "NBG-051",
      "name": "def_isCard",
      "kind": "Definitional",
      "statement_unicode": "∀(C:Class), ((Card(C) → ∃(K:Class), (CardOf(K) = C)) ∧ (∃(K:Class), (CardOf(K) = C) → Card(C)))",
      "proof_unicode": "plan_card_def",
      "statement_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))))"
    },
    {
      "id": 52,
      "num": "NBG-052",
      "name": "thm_omega_card",
      "kind": "Derived",
      "statement_unicode": "Card(ω)",
      "proof_unicode": "(snd((plan_card_def ω)) ⟨ω, ax_cardof_omega⟩)",
      "statement_ast": "(Card omega)"
    },
    {
      "id": 53,
      "num": "NBG-053",
      "name": "thm_on_card",
      "kind": "Derived",
      "statement_unicode": "Card(On)",
      "proof_unicode": "(snd((plan_card_def On)) ⟨On, ax_cardof_on⟩)",
      "statement_ast": "(Card On)"
    },
    {
      "id": 54,
      "num": "NBG-054",
      "name": "thm_v_not_card",
      "kind": "Derived",
      "statement_unicode": "¬Card(V)",
      "proof_unicode": "(λ(hCardV:Card(V)). ((ax_cardof_never_v fst((fst((plan_card_def V)) hCardV))) snd((fst((plan_card_def V)) hCardV))))",
      "statement_ast": "(Pi (_ : (Card V)) -> False)"
    },
    {
      "id": 55,
      "num": "NBG-055",
      "name": "ax_tarski_universes",
      "kind": "Axiomatic",
      "statement_unicode": "∀(X:Class), (set(X) → ∃(U:Class), (universe(U) ∧ ((X ∈ U) ∧ set(U))))",
      "proof_unicode": "plan_tarski_axm",
      "statement_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))))))"
    },
    {
      "id": 56,
      "num": "NBG-056",
      "name": "def_binary_union",
      "kind": "Definitional",
      "statement_unicode": "∀(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))))",
      "proof_unicode": "plan_bin_union",
      "statement_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))))))"
    },
    {
      "id": 57,
      "num": "NBG-057",
      "name": "def_binary_inter",
      "kind": "Definitional",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ∀(z:Class), (((z ∈ (A ∩ B)) → ((z ∈ A) ∧ (z ∈ B))) ∧ (((z ∈ A) ∧ (z ∈ B)) → (z ∈ (A ∩ B))))",
      "proof_unicode": "plan_bin_inter",
      "statement_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))))))"
    },
    {
      "id": 58,
      "num": "NBG-058",
      "name": "def_unary_union",
      "kind": "Definitional",
      "statement_unicode": "∀(A:Class), ∀(z:Class), (((z ∈ ⋃(A)) → ∃(y:Class), ((y ∈ A) ∧ (z ∈ y))) ∧ (∃(y:Class), ((y ∈ A) ∧ (z ∈ y)) → (z ∈ ⋃(A))))",
      "proof_unicode": "plan_unary_union",
      "statement_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)))))"
    },
    {
      "id": 59,
      "num": "NBG-059",
      "name": "def_unary_inter",
      "kind": "Definitional",
      "statement_unicode": "∀(A:Class), ∀(z:Class), (((z ∈ ⋂(A)) → ∀(y:Class), ((y ∈ A) → (z ∈ y))) ∧ (∀(y:Class), ((y ∈ A) → (z ∈ y)) → (z ∈ ⋂(A))))",
      "proof_unicode": "plan_unary_inter",
      "statement_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)))))"
    },
    {
      "id": 60,
      "num": "NBG-060",
      "name": "def_singleton",
      "kind": "Definitional",
      "statement_unicode": "∀(a:Class), ∀(z:Class), (((z ∈ {a}) → (z = a)) ∧ ((z = a) → (z ∈ {a})))",
      "proof_unicode": "plan_singleton",
      "statement_ast": "(Pi (a : Class) -> (Pi (z : Class) -> (Sigma (_ : (Pi (_ : (Id Prop ((In z) (Singleton a)) true)) -> (Id Class z a))) * (Pi (_ : (Id Class z a)) -> (Id Prop ((In z) (Singleton a)) true)))))"
    },
    {
      "id": 61,
      "num": "NBG-061",
      "name": "def_upair",
      "kind": "Definitional",
      "statement_unicode": "∀(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})))",
      "proof_unicode": "plan_upair",
      "statement_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))))))"
    },
    {
      "id": 62,
      "num": "NBG-062",
      "name": "thm_russell_class_not_set",
      "kind": "Derived",
      "statement_unicode": "¬∃(R:Class), (set(R) ∧ ∀(x:Class), (((x ∈ R) → (set(x) ∧ ¬(x ∈ x))) ∧ ((set(x) ∧ ¬(x ∈ x)) → (x ∈ R))))",
      "proof_unicode": "(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))⟩)))",
      "statement_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)"
    },
    {
      "id": 63,
      "num": "NBG-063",
      "name": "thm_dunaev_tagged_pair_proj",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ((pr1((A, B)) = A) ∧ (pr2((A, B)) = B))",
      "proof_unicode": "(λ(A:Class). (λ(B:Class). ⟨((ax_dunaev_pr1 A) B), ((ax_dunaev_pr2 A) B)⟩))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Sigma (_ : (Id Class (DunaevPr1 ((DunaevPair A) B)) A)) * (Id Class (DunaevPr2 ((DunaevPair A) B)) B))))"
    },
    {
      "id": 64,
      "num": "NBG-064",
      "name": "thm_dunaev_pair_injectivity",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ∀(C:Class), ∀(D:Class), (((A, B) = (C, D)) → ((A = C) ∧ (B = D)))",
      "proof_unicode": "(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). (λ(B:Class). ⟨((ax_dunaev_pr1 A) B), ((ax_dunaev_pr2 A) B)⟩)) C) D)))⟩)))))",
      "statement_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)))))))"
    },
    {
      "id": 65,
      "num": "NBG-065",
      "name": "thm_empty_set_elim",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), (∀(z:Class), ¬(z ∈ X) → (X = ∅))",
      "proof_unicode": "(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)))⟩))))",
      "statement_ast": "(Pi (X : Class) -> (Pi (_ : (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) X) true)) -> False))) -> (Id Class X EmptySet)))"
    },
    {
      "id": 66,
      "num": "NBG-066",
      "name": "thm_every_set_in_v",
      "kind": "Derived",
      "statement_unicode": "∀(x:Class), (set(x) → (x ∈ V))",
      "proof_unicode": "(λ(x:Class). snd(((fst((plan_universal_class_def V)) refl(V)) x)))",
      "statement_ast": "(Pi (x : Class) -> (Pi (_ : (Id Prop (Set x) true)) -> (Id Prop ((In x) V) true)))"
    },
    {
      "id": 67,
      "num": "NBG-067",
      "name": "thm_in_v_is_set",
      "kind": "Derived",
      "statement_unicode": "∀(x:Class), ((x ∈ V) → set(x))",
      "proof_unicode": "(λ(x:Class). fst(((fst((plan_universal_class_def V)) refl(V)) x)))",
      "statement_ast": "(Pi (x : Class) -> (Pi (_ : (Id Prop ((In x) V) true)) -> (Id Prop (Set x) true)))"
    },
    {
      "id": 68,
      "num": "NBG-068",
      "name": "thm_emptyset_in_v",
      "kind": "Derived",
      "statement_unicode": "(∅ ∈ V)",
      "proof_unicode": "(((λ(x:Class). snd(((fst((plan_universal_class_def V)) refl(V)) x))) ∅) ax_empty_set)",
      "statement_ast": "(Id Prop ((In EmptySet) V) true)"
    },
    {
      "id": 69,
      "num": "NBG-069",
      "name": "thm_omega_in_v",
      "kind": "Derived",
      "statement_unicode": "(ω ∈ V)",
      "proof_unicode": "(((λ(x:Class). snd(((fst((plan_universal_class_def V)) refl(V)) x))) ω) plan_omega_set)",
      "statement_ast": "(Id Prop ((In omega) V) true)"
    },
    {
      "id": 70,
      "num": "NBG-070",
      "name": "thm_power_omega_is_set",
      "kind": "Derived",
      "statement_unicode": "set(𝒫(ω))",
      "proof_unicode": "((plan_exponent ω) plan_omega_set)",
      "statement_ast": "(Id Prop (Set (Power omega)) true)"
    },
    {
      "id": 71,
      "num": "NBG-071",
      "name": "thm_singleton_self_in",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), (a ∈ {a})",
      "proof_unicode": "(λ(a:Class). (snd(((plan_singleton a) a)) refl(a)))",
      "statement_ast": "(Pi (a : Class) -> (Id Prop ((In a) (Singleton a)) true))"
    },
    {
      "id": 72,
      "num": "NBG-072",
      "name": "thm_upair_left_in",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), ∀(b:Class), (a ∈ {a, b})",
      "proof_unicode": "(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))))))))",
      "statement_ast": "(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((In a) ((UPair a) b)) true)))"
    },
    {
      "id": 73,
      "num": "NBG-073",
      "name": "thm_upair_right_in",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), ∀(b:Class), (b ∈ {a, b})",
      "proof_unicode": "(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))))))))",
      "statement_ast": "(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((In b) ((UPair a) b)) true)))"
    },
    {
      "id": 74,
      "num": "NBG-074",
      "name": "thm_inter_subset_left",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ((A ∩ B) ⊆ A)",
      "proof_unicode": "(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)))))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) A) true)))"
    },
    {
      "id": 75,
      "num": "NBG-075",
      "name": "thm_inter_subset_right",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ((A ∩ B) ⊆ B)",
      "proof_unicode": "(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)))))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) B) true)))"
    },
    {
      "id": 76,
      "num": "NBG-076",
      "name": "thm_union_subset_left",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (A ⊆ (A ∪ B))",
      "proof_unicode": "(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))))))))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset A) ((BinUnion A) B)) true)))"
    },
    {
      "id": 77,
      "num": "NBG-077",
      "name": "thm_union_subset_right",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (B ⊆ (A ∪ B))",
      "proof_unicode": "(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))))))))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset B) ((BinUnion A) B)) true)))"
    },
    {
      "id": 78,
      "num": "NBG-078",
      "name": "thm_inter_idem_subset",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ((A ∩ A) ⊆ A)",
      "proof_unicode": "(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))",
      "statement_ast": "(Pi (A : Class) -> (Id Prop ((Subset ((BinInter A) A)) A) true))"
    },
    {
      "id": 79,
      "num": "NBG-079",
      "name": "thm_union_idem_subset",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), (A ⊆ (A ∪ A))",
      "proof_unicode": "(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))",
      "statement_ast": "(Pi (A : Class) -> (Id Prop ((Subset A) ((BinUnion A) A)) true))"
    },
    {
      "id": 80,
      "num": "NBG-080",
      "name": "thm_inter_set_left",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (set(A) → set((A ∩ B)))",
      "proof_unicode": "(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⟩))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Pi (_ : (Id Prop (Set A) true)) -> (Id Prop (Set ((BinInter A) B)) true))))"
    },
    {
      "id": 81,
      "num": "NBG-081",
      "name": "thm_semi_singleton_sub_char",
      "kind": "Derived",
      "statement_unicode": "∀(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))))",
      "proof_unicode": "(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)⟩))))))",
      "statement_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))))))))"
    },
    {
      "id": 82,
      "num": "NBG-082",
      "name": "thm_family_value_eval",
      "kind": "Derived",
      "statement_unicode": "∀(F:Class), ∀(x:Class), ∀(z:Class), ∀(s:Class), ∀(u:Class), (((s ⊆ z) ∧ ((u ∈ ((ValOf F) x)) → (u ∈ s))) → ((u ∈ ((ValOf F) x)) → (u ∈ z)))",
      "proof_unicode": "(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)))))))))",
      "statement_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))))))))"
    },
    {
      "id": 83,
      "num": "NBG-083",
      "name": "thm_sigma_model_proj",
      "kind": "Derived",
      "statement_unicode": "∀(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))))",
      "proof_unicode": "(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))))))",
      "statement_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))))))"
    },
    {
      "id": 84,
      "num": "NBG-084",
      "name": "thm_pi_model_elim",
      "kind": "Derived",
      "statement_unicode": "∀(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)))",
      "proof_unicode": "(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))))))",
      "statement_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)))))"
    },
    {
      "id": 85,
      "num": "NBG-085",
      "name": "thm_if_dep_true_branch",
      "kind": "Derived",
      "statement_unicode": "∀(Phi:Prop), ∀(T1:(Phi → Prop)), ∀(T2:(¬Phi → Prop)), ((∀(h:Phi), (T1 h) ∧ ∀(nh:¬Phi), (T2 nh)) → ∀(h:Phi), (T1 h))",
      "proof_unicode": "(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))))))",
      "statement_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))))))"
    },
    {
      "id": 86,
      "num": "NBG-086",
      "name": "thm_if_dep_false_branch",
      "kind": "Derived",
      "statement_unicode": "∀(Phi:Prop), ∀(T1:(Phi → Prop)), ∀(T2:(¬Phi → Prop)), ((∀(h:Phi), (T1 h) ∧ ∀(nh:¬Phi), (T2 nh)) → ∀(nh:¬Phi), (T2 nh))",
      "proof_unicode": "(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))))))",
      "statement_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))))))"
    },
    {
      "id": 87,
      "num": "NBG-087",
      "name": "thm_model_stratification_trans",
      "kind": "Derived",
      "statement_unicode": "∀(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)))",
      "proof_unicode": "(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)))))))))))",
      "statement_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))))))"
    },
    {
      "id": 88,
      "num": "NBG-088",
      "name": "thm_dunaev_pair_fst_beta",
      "kind": "Derived",
      "statement_unicode": "∀(x:Class), ∀(y:Class), (pr1((x, y)) = x)",
      "proof_unicode": "(λ(x:Class). (λ(y:Class). ((ax_dunaev_pr1 x) y)))",
      "statement_ast": "(Pi (x : Class) -> (Pi (y : Class) -> (Id Class (DunaevPr1 ((DunaevPair x) y)) x)))"
    },
    {
      "id": 89,
      "num": "NBG-089",
      "name": "thm_dunaev_pair_snd_beta",
      "kind": "Derived",
      "statement_unicode": "∀(x:Class), ∀(y:Class), (pr2((x, y)) = y)",
      "proof_unicode": "(λ(x:Class). (λ(y:Class). ((ax_dunaev_pr2 x) y)))",
      "statement_ast": "(Pi (x : Class) -> (Pi (y : Class) -> (Id Class (DunaevPr2 ((DunaevPair x) y)) y)))"
    },
    {
      "id": 90,
      "num": "NBG-090",
      "name": "thm_singleton_elem_eq",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), ∀(z:Class), ((z ∈ {a}) → (z = a))",
      "proof_unicode": "(λ(a:Class). (λ(z:Class). (λ(hz:(z ∈ {a})). (fst(((plan_singleton a) z)) hz))))",
      "statement_ast": "(Pi (a : Class) -> (Pi (z : Class) -> (Pi (_ : (Id Prop ((In z) (Singleton a)) true)) -> (Id Class z a))))"
    },
    {
      "id": 91,
      "num": "NBG-091",
      "name": "thm_singleton_subset_upair_left",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), ∀(b:Class), ({a} ⊆ {a, b})",
      "proof_unicode": "(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})). (fst(((plan_singleton a) z)) hz)))) a) z) hz)))))))))))",
      "statement_ast": "(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((Subset (Singleton a)) ((UPair a) b)) true)))"
    },
    {
      "id": 92,
      "num": "NBG-092",
      "name": "thm_singleton_subset_upair_right",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), ∀(b:Class), ({b} ⊆ {a, b})",
      "proof_unicode": "(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})). (fst(((plan_singleton a) z)) hz)))) b) z) hz)))))))))))",
      "statement_ast": "(Pi (a : Class) -> (Pi (b : Class) -> (Id Prop ((Subset (Singleton b)) ((UPair a) b)) true)))"
    },
    {
      "id": 93,
      "num": "NBG-093",
      "name": "thm_inter_subset_union",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), ((A ∩ B) ⊆ (A ∪ B))",
      "proof_unicode": "(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))))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Id Prop ((Subset ((BinInter A) B)) ((BinUnion A) B)) true)))"
    },
    {
      "id": 94,
      "num": "NBG-094",
      "name": "thm_inter_set_right",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (set(B) → set((A ∩ B)))",
      "proof_unicode": "(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⟩))))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Pi (_ : (Id Prop (Set B) true)) -> (Id Prop (Set ((BinInter A) B)) true))))"
    },
    {
      "id": 95,
      "num": "NBG-095",
      "name": "thm_emptyset_subset_all",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), (∅ ⊆ X)",
      "proof_unicode": "(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)))",
      "statement_ast": "(Pi (X : Class) -> (Id Prop ((Subset EmptySet) X) true))"
    },
    {
      "id": 96,
      "num": "NBG-096",
      "name": "thm_subset_antisym",
      "kind": "Derived",
      "statement_unicode": "∀(X:Class), ∀(Y:Class), (((X ⊆ Y) ∧ (Y ⊆ X)) → (X = Y))",
      "proof_unicode": "(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)⟩)))))",
      "statement_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))))"
    },
    {
      "id": 97,
      "num": "NBG-097",
      "name": "thm_every_set_subset_v",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), (∀(z:Class), ((z ∈ A) → set(z)) → (A ⊆ V))",
      "proof_unicode": "(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)))))))",
      "statement_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)))"
    },
    {
      "id": 98,
      "num": "NBG-098",
      "name": "thm_singleton_in_v_if_set",
      "kind": "Derived",
      "statement_unicode": "∀(a:Class), (set({a}) → ({a} ∈ V))",
      "proof_unicode": "(λ(a:Class). (λ(hSetSing:set({a})). (((λ(x:Class). snd(((fst((plan_universal_class_def V)) refl(V)) x))) {a}) hSetSing)))",
      "statement_ast": "(Pi (a : Class) -> (Pi (_ : (Id Prop (Set (Singleton a)) true)) -> (Id Prop ((In (Singleton a)) V) true)))"
    },
    {
      "id": 99,
      "num": "NBG-099",
      "name": "thm_power_omega_in_v",
      "kind": "Derived",
      "statement_unicode": "(𝒫(ω) ∈ V)",
      "proof_unicode": "(((λ(x:Class). snd(((fst((plan_universal_class_def V)) refl(V)) x))) 𝒫(ω)) ((plan_exponent ω) plan_omega_set))",
      "statement_ast": "(Id Prop ((In (Power omega)) V) true)"
    },
    {
      "id": 100,
      "num": "NBG-100",
      "name": "thm_inter_subset_both",
      "kind": "Derived",
      "statement_unicode": "∀(A:Class), ∀(B:Class), (((A ∩ B) ⊆ A) ∧ ((A ∩ B) ⊆ B))",
      "proof_unicode": "(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)⟩))",
      "statement_ast": "(Pi (A : Class) -> (Pi (B : Class) -> (Sigma (_ : (Id Prop ((Subset ((BinInter A) B)) A) true)) * (Id Prop ((Subset ((BinInter A) B)) B) true))))"
    }
  ]
}