Cartesian Product is Small iff Inverse is Small

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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.

$\blacksquare$


Sources