# Cartesian Product is Small iff Inverse is Small

## Theorem

Let $A$ and $B$ be classes.

Then the Cartesian product $A \times B$ is a small class if and only if $B \times A$ is small.

## Proof

 $\displaystyle A \times B$ $=$ $\displaystyle \left\{ { \left({ x, y }\right) : x \in A \land y \in B }\right\}$ Definition of Cartesian Product $\displaystyle$ $=$ $\displaystyle \left\{ { \left({ y, x }\right) : x \in A \land y \in B }\right\}^{-1}$ Definition of Inverse Relation $\displaystyle$ $=$ $\displaystyle \left({ B \times A }\right)^{-1}$ Definition of Cartesian Product

Let $B \times A$ be a small class.

Then, by Inverse of Small Relation is Small, $A \times B$ is also small.

Similarly, let $A \times B$ be small.

Then, by Inverse of Small Relation is Small, $B \times A$ is also small.

$\blacksquare$