Union of Doubleton

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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

$\blacksquare$


Sources