Union is Idempotent

Theorem

$S \cup S = S$

Proof

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

$\blacksquare$