# Union is Idempotent

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## Theorem

$S \cup S = S$

## Proof

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

$\blacksquare$