# Union of Doubleton

## Theorem

Let $x$ and $y$ be sets.

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

Then $\ds \bigcup \set {x, y}$ is a set such that:

$\ds \bigcup \set {x, y} = x \cup y$

## Proof

 $\ds$  $\ds z \in \bigcup \set {x, y}$ $\ds \leadstoandfrom \ \$ $\ds$  $\ds \exists w \in \set {x, y}: z \in w$ Definition of Union of Class $\ds \leadstoandfrom \ \$ $\ds$  $\ds \exists w: \paren {\paren {w = x \lor w = y} \land z \in w}$ Definition of Doubleton Class $\ds \leadstoandfrom \ \$ $\ds$  $\ds \paren {z \in x \lor z \in y}$ Equality implies Substitution $\ds \leadstoandfrom \ \$ $\ds$  $\ds z \in \paren {x \cup y}$ Definition of Class Union

Then, from Axiom of Unions, it follows that $x \cup y$ is a set.

$\blacksquare$