Union of Disjoint Singletons is Doubleton

Theorem
Let $\left\{{a}\right\}$ and $\left\{{b}\right\}$ be singletons such that $a \ne b$.

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

Proof
Let $x \in \left\{{a}\right\} \cup \left\{{b}\right\}$.

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

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

Thus by definition of subset:


 * $\left\{{a}\right\} \cup \left\{{b}\right\} \subseteq \left\{{a, b}\right\}$

Let $x \in \left\{{a, b}\right\}$.

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

So by the Axiom of Union:
 * $x \in \left\{{a}\right\} \cup \left\{{b}\right\}$

Thus by definition of subset:


 * $\left\{{a, b}\right\} \subseteq \left\{{a}\right\} \cup \left\{{b}\right\}$

The result follows by definition of set equality.