Union of Disjoint Singletons is Doubleton

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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:

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

$\Box$


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:

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

$\Box$


The result follows by definition of set equality.

$\blacksquare$


Sources