Union of Small Classes is Small

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.

By the axiom of pairing:


 * $\mathscr M \left({\left\{{x, y}\right\}}\right)$

By the axiom of union:


 * $\mathscr M \left({\bigcup \left\{{x, y}\right\}}\right)$

By Union of Doubleton:


 * $\mathscr M \left({x \cup y}\right)$