Condition for Cartesian Product Equivalent to Associated Cardinal Number

Theorem
Let $S$ and $T$ be nonempty sets.

Then:


 * $S \times T \sim | S \times T | \iff S \sim | S | \land T \sim | T |$

Necessary Condition
If $S \times T \sim | S \times T |$, then there is a mapping $f$ such that:


 * $f : S \times T \to | S \times T |$ is a bijection.

Since $f$ is a bijection, it follows that:


 * $S$ is equivalent to the image of $S \times \{ x \}$ under $f$ where $x \in T$.

This, in turn, is a subset of the ordinal $| S \times T |$.

$| S \times T |$ is an ordinal by Cardinal Number is Ordinal Number.

By Condition for Set Equivalent to Cardinal Number, it follows that $S \sim |S|$.

Similarly, $T \sim |T|$.

Sufficient Condition
Suppose $f : S \to |S|$ is a bijection and $g : T \to |T|$ is a bijection.

Define the function $F$ to be:


 * $F \langle x, y \rangle = |S| g\left({ y }\right) + f \left({ x }\right)$ if $x \in S,y \in T$.

It follows that $F : S \times T \to |S| |T|$ is a injection.

By Condition for Set Equivalent to Cardinal Number, it follows that $S \times T \sim |S \times T|$.