Class Equality Extension of Set Equality

Theorem

Let $=_s$ denote set equality.

Let $=_c$ denote class equality.

Let $x$ and $y$ be sets.

Then $x =_s y$ if and only if $x =_c y$.

Proof

\(\ds x\)

\(=_s\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall z: \, \)

\(\ds \leftparen {z \in x}\)

\(\iff\)

\(\ds \rightparen {z \in y}\)

Definition of Set Equality

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=_c\)

\(\ds y\)

Definition of Class Equality, Class Membership is Extension of Set Membership

$\blacksquare$

The validity of the material on this page is questionable. In particular: This has been put in the category Category:Zermelo-Fraenkel Class Theory. Should it really be that specific? Does it not also apply to Definition:NBG for example? Or others? Complete review of the applicability of this is of high importance, because I believe that upon this depends the applicability of identifying a small class with a set. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.