Class Equality Extension of Set Equality

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Let $=_s$ denote set equality.

Let $=_c$ denote class equality.

Let $x$ and $y$ be sets.

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


\(\ds x\) \(=_s\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall z: \, \) \(\ds \leftparen {z \in x}\) \(\iff\) \(\ds \rightparen {z \in y}\) Definition of Set Equality
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=_c\) \(\ds y\) Definition of Class Equality, Class Membership is Extension of Set Membership