# Union of Doubleton

## Theorem

Let $\left\{{x, y}\right\}$ be a doubleton.

Then:

$\displaystyle \bigcup \left\{{x, y}\right\} = x \cup y$

## Proof

 $\displaystyle z \in \bigcup \left\{ {x, y}\right\}$ $\iff$ $\displaystyle \exists w \in \left\{ {x, y}\right\}: z \in w$ Definition of Union of Set of Sets $\displaystyle$ $\iff$ $\displaystyle \exists w: \left({\left({w = x \lor w = y}\right) \land z \in w}\right)$ Definition of Doubleton $\displaystyle$ $\iff$ $\displaystyle \left({z \in x \lor z \in y}\right)$ Equality implies Substitution $\displaystyle$ $\iff$ $\displaystyle z \in \left({x \cup y}\right)$ Definition of Set Union

$\blacksquare$