# Class Equality Extension of Set Equality

## 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

$\blacksquare$