Class Equality Extension of Set Equality

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Theorem

Let $=_1$ denote set equality.

Let $=_2$ denote class equality.

Let $x$ and $y$ be sets.


Then $x =_1 y$ iff $x =_2 y$.


Proof

\(\displaystyle x =_1 y\) \(\iff\) \(\displaystyle \forall z: \left({ z \in x \iff z \in y }\right)\) Definition of set equality
\(\displaystyle \) \(\iff\) \(\displaystyle x =_2 y\) Definition of class equality, Class Membership Extension of Set Membership

$\blacksquare$