Union of Finite Sets is Finite

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

Then $S \cup T$ is a finite set.

Proof
Note that $\left|{ S \cup T }\right| \le \left|{ S \times T }\right|$ by Cardinal of Union Less than Cardinal of Cartesian Product.

The theorem follows from the fact that $S \times T$ is finite by Product of Finite Sets is Finite.