Union of Small Classes is Small

Theorem

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

Then $x \cup y$ is also small.

Proof

Let $\map {\mathscr M} A$ denote that $A$ is small.

 $\ds \map {\mathscr M} x \land \map {\mathscr M} y$ $\leadsto$ $\ds \map {\mathscr M} {\set {x, y} }$ Axiom of Pairing $\ds$ $\leadsto$ $\ds \map {\mathscr M} {\bigcup \set {x, y} }$ Axiom of Unions $\ds$ $\leadsto$ $\ds \map {\mathscr M} {x \cup y}$ Union of Doubleton

$\blacksquare$