Unordered Pair is Finite

Theorem
Let $x, y$ be arbitrary.

Then $\left\{ {x, y}\right\}$ is finite.

Proof
By Pair is Union of Singletons:
 * $\left\{ {x, y}\right\} = \left\{ {x}\right\} \cup \left\{ {y}\right\}$

By Singleton is Finite:
 * $\left\{ {x}\right\}$ and $\left\{ {y}\right\}$ are finite.

Thus by Union of Finite Sets is Finite:
 * $\left\{ {x, y}\right\}$ is finite.