# Union is Idempotent

## Theorem

$S \cup S = S$

## Proof

 $\displaystyle x$ $\in$ $\displaystyle S \cup S$ $\quad$ $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \in S$ $\lor$ $\displaystyle x \in S$ $\quad$ Definition of Set Union $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle S$ $\quad$ Rule of Idempotence: Disjunction $\quad$

$\blacksquare$