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

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 an equivalent to the axiom of subsets, $\left({ a \times b }\right)$ is small.