Union of Finite Sets is Finite/Proof 2

Theorem

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

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

Proof

Note that $\card {S \cup T} \le \card {S \times T}$ 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.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.29 \ (1)$