Union of Disjoint Singletons is Doubleton

Theorem
Let $\set a$ and $\set b$ be singletons such that $a \ne b$.

Then:
 * $\set a \cup \set b = \set {a, b}$

Proof
Let $x \in \set a \cup \set b$.

Then by the Axiom of Unions:
 * $x = a \lor x = b$

It follows from the Axiom of Pairing that:
 * $x \in \set {a, b}$

Thus by definition of subset:


 * $\set a \cup \set b \subseteq \set {a, b}$

Let $x \in \set {a, b}$.

Then by the Axiom of Pairing:
 * $x = a \lor x = b$

So by the Axiom of Unions:
 * $x \in \set a \cup \set b$

Thus by definition of subset:


 * $\set {a, b} \subseteq \set a \cup \set b$

The result follows by definition of set equality.