Unordered Pair is Finite

Theorem
Let $x, y$ be arbitrary.

Then $\set {x, y}$ is finite.

Proof
By Union of Disjoint Singletons is Doubleton:
 * $\set {x, y} = \set x \cup \set y$

By Singleton is Finite:
 * $\set x$ and $\set y$ are finite.

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