Union of Finite Sets is Finite/Proof 2
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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: $\S 10.29 \ (1)$