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 unions:


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

By Union of Doubleton:


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