# Union of Small Classes is Small

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## Theorem

Let $x$ and $y$ be small classes.

Then $x \cup y$ is also small.

## Proof

Let $\mathscr M \left({A}\right)$ denote that $A$ is small.

 $\displaystyle \mathscr M \left({x}\right) \land \mathscr M \left({y}\right)$ $\implies$ $\displaystyle \mathscr M \left({\left\{{x, y}\right\}}\right)$ Axiom of Pairing $\displaystyle$ $\implies$ $\displaystyle \mathscr M \left({\bigcup \left\{{x, y}\right\}}\right)$ Axiom of Union $\displaystyle$ $\implies$ $\displaystyle \mathscr M \left({x \cup y}\right)$ Union of Doubleton

$\blacksquare$