DefClassConditional (3.22, Definition): IF-THEN-ELSE-FI (Class conditional with NFV side condition and context turnstile)

Standalone kernel entity record from 3. Set/Class-Theoretic Foundations (3.01–3.78) — View in Complete Entity Database (#DefClassConditional) →

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DefClassConditional (3.22) — IF-THEN-ELSE-FI (Class conditional with NFV side condition and context turnstile)

Morse-Kelley / Von Neumann-Bernays-Gödel class comprehension, sethood, ordinals, Dunaev pairs, cardinals, universes, and reflection.

Num Entity Database Record (Vertical Field Table)
3.22
Title IF-THEN-ELSE-FI (Class conditional with NFV side condition and context turnstile) #DefClassConditional #label-if-then-else-fi-class-conditional-with-nfv-side-condition-and-context-turnstile
Classification Definition
Formula
0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\Gamma} \href{../symbols/typing-colon.html}{:} \href{../symbols/ctx-sort.html}{\mathsf{Ctx}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{v} \href{../symbols/typing-colon.html}{:} \href{../symbols/variable-sort.html}{\mathsf{Variable}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\psi}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \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/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/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/dot-delim.html}{.} \href{../symbols/ifv-pred.html}{\mathrm{IFV}} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{v} \href{../symbols/comma-delim.html}{,} \href{../symbols/formula-metavariables.html}{\boldsymbol{\psi}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/theory-wau.html}{\mathcal{W}} \href{../symbols/comma-delim.html}{,} \href{../symbols/formula-metavariables.html}{\Gamma} \href{../symbols/turnstile-derivability.html}{\vdash} \href{../symbols/quotation.html}{\langle } \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/class-if-then-else.html}{\mathbf{IF}} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\psi}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/class-if-then-else.html}{\mathbf{THEN}} \href{../symbols/variables.html}{A} \href{../symbols/class-if-then-else.html}{\mathbf{ELSE}} \href{../symbols/variables.html}{B} \href{../symbols/class-if-then-else.html}{\mathbf{FI}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/eq-pred.html}{=} \href{../symbols/class-comprehension.html}{\{} \href{../symbols/binder-brackets.html}{\{\!\lvert } \href{../symbols/variables.html}{v} \href{../symbols/binder-brackets.html}{\rvert\!\}} \href{../symbols/class-comprehension.html}{ \mid } \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\psi}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/and-conn.html}{\land} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/binder-brackets.html}{\{\!\lvert } \href{../symbols/variables.html}{v} \href{../symbols/binder-brackets.html}{\rvert\!\}} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{A} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/or-conn.html}{\lor} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\psi}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/and-conn.html}{\land} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/binder-brackets.html}{\{\!\lvert } \href{../symbols/variables.html}{v} \href{../symbols/binder-brackets.html}{\rvert\!\}} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{B} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/class-comprehension.html}{\}} \href{../symbols/quotation.html}{\rangle } \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST CpIf
Scheme of Types 0(∀([Γ@Variable]:[Ctx@Type])∀([v@Variable]:[Variable@Type])∀([𝛙@Variable]:[Fm@Type])∀([A@Variable]:[Class@Type])∀([B@Variable]:[Class@Type])((¬Fm.IFV([v@Variable.Expression],[𝛙@Fm.Expression]))→(𝘞,[Γ@Ctx.Expression]⊢‹⟦IF ⟦[𝛙@Fm.Expression]⟧ THEN [A@Class.Expression] ELSE [B@Class.Expression] FI⟧={[⦃[v@Variable.Expression]⦄@Class.Expression]|(⟦[𝛙@Fm.Expression]⟧∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[A@Class.Expression]))∨((¬⟦[𝛙@Fm.Expression]⟧)∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[B@Class.Expression]))}›)))
Verification Definitional (Kernel AST) · Standalone Entry (DefClassConditional) → · (3.22) · #DefClassConditional