# Class Equality Extension of Set Equality

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## 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.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. |