AxmRightInSound (3.09, Axiom): “Membership respects equality from the right”

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

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Proof Trees:

AxmRightInSound (3.09) — “Membership respects equality from the right”

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

Num Entity Database Record (Vertical Field Table)
3.09
Title “Membership respects equality from the right” #AxmRightInSound #label-membership-respects-equality-from-the-right
Classification Axiom
Formula
0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{X} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{Y} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{X} \href{../symbols/eq-pred.html}{=} \href{../symbols/variables.html}{Y} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/forall-binder.html}{\forall} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{W} \href{../symbols/typing-colon.html}{:} \href{../symbols/class-type.html}{\mathsf{Class}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{X} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{W} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/iff-conn.html}{\longleftrightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/variables.html}{Y} \href{../symbols/in-pred.html}{\in} \href{../symbols/variables.html}{W} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST PRightInSoundAx_prf
Scheme of Types 0(∀([X@Variable]:[Class@Type])∀([Y@Variable]:[Class@Type])(([X@Class.Expression]=[Y@Class.Expression])→∀([W@Variable]:[Class@Type])(([X@Class.Expression]∈[W@Class.Expression])⟷([Y@Class.Expression]∈[W@Class.Expression]))))
Verification Postulated (Kernel Axiom) · Standalone Entry (AxmRightInSound) → · (3.09) · #AxmRightInSound