DefPairfulClass (3.46, Definition): “Is pairful” predicate

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

Formula Display:
Proof Trees:

DefPairfulClass (3.46) — “Is pairful” predicate

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

Num Entity Database Record (Vertical Field Table)
3.46
Title “Is pairful” predicate #DefPairfulClass #label-is-pairful-predicate
Classification Definition
Formula
0\(\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/delimiters-brackets.html}{(} \href{../symbols/pairful-pred.html}{\mathsf{Pairful}} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/iff-conn.html}{\longleftrightarrow} \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/delimiters-brackets.html}{(} \href{../symbols/variables.html}{A} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/and-conn.html}{\land} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{B} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{U} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \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/in-pred.html}{\in} \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/delimiters-brackets.html}{)} \)
SML Kernel AST PKernelDef_prf
Scheme of Types 0(∀([U@Variable]:[Class@Type])(Pairful([U@Class.Expression])⟷(∀([A@Variable]:[Class@Type])∀([B@Variable]:[Class@Type])((([A@Class.Expression]∈[U@Class.Expression])∧([B@Class.Expression]∈[U@Class.Expression]))→({[A@Class.Expression],[B@Class.Expression]}∈[U@Class.Expression])))))
Verification Definitional (Kernel AST) · Standalone Entry (DefPairfulClass) → · (3.46) · #DefPairfulClass