DefIsSingleton (3.27, Definition): “Has exactly one element” predicate

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

Formula Display:
Proof Trees:

DefIsSingleton (3.27) — “Has exactly one element” predicate

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

Num Entity Database Record (Vertical Field Table)
3.27
Title “Has exactly one element” predicate #DefIsSingleton #label-has-exactly-one-element-predicate
Classification Definition
Formula
0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{s} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{s} \href{../symbols/singleton-pred.html}{\mathsf{singleton}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/iff-conn.html}{\longleftrightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/exists-uniq-binder.html}{\exists!} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{x} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{x} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{s} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST TpSingleton
Scheme of Types 0(∀([s@Variable]:[Class@Type])(([s@Class.Expression] singleton)⟷(∃!([x@Variable]:[Class@Type])([x@Class.Expression]∈[s@Class.Expression]))))
Verification Definitional (Kernel AST) · Standalone Entry (DefIsSingleton) → · (3.27) · #DefIsSingleton