Cartesian Product is Small

Theorem
Let $a$ and $b$ be small classes.

Then their Cartesian product $\left({a \times b}\right)$ is small:


 * $\mathscr M \left({a \times b}\right)$

Proof
So by the definition of power set:
 * $\left({a \times b}\right) \subseteq \mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$

By Union of Small Classes is Small, $\left({a \cup b}\right)$ is small.

By the Axiom of Powers, $\mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$ is small.

By Axiom of Subsets Equivalents, $\left({a \times b}\right)$ is small.