AxmLeftInSound (3.08, Axiom): “Membership respects equality from the left”

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

Formula Display:
Proof Trees:

AxmLeftInSound (3.08) — “Membership respects equality from the left”

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

Num Entity Database Record (Vertical Field Table)
3.08
Title “Membership respects equality from the left” #AxmLeftInSound #label-membership-respects-equality-from-the-left
Classification Axiom
Formula
0(∀(X:Class)∀(Y:Class)((X=Y)→∀(z:Class)((z∈X)⟷(z∈Y))))
0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{X} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{Y} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{X} \href{../symbols/eq-pred.html}{=} \href{../symbols/variables.html}{Y} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{z} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{z} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{X} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/iff-conn.html}{\longleftrightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{z} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{Y} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST PLeftInSoundAx_prf
Scheme of Types 0(∀([X@Variable]:[Class@Type])∀([Y@Variable]:[Class@Type])(([X@Class.Expression]=[Y@Class.Expression])→∀([z@Variable]:[Class@Type])(([z@Class.Expression]∈[X@Class.Expression])⟷([z@Class.Expression]∈[Y@Class.Expression]))))
Verification Postulated (Kernel Axiom) · Standalone Entry (AxmLeftInSound) → · (3.08) · #AxmLeftInSound