Union is Commutative

Theorem

$S \cup T = T \cup S$

Proof

 $\displaystyle x$ $\in$ $\displaystyle \paren {S \cup T}$ $\quad$ $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \in S$ $\lor$ $\displaystyle x \in T$ $\quad$ Definition of Set Union $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \in T$ $\lor$ $\displaystyle x \in S$ $\quad$ Disjunction is Commutative $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {T \cup S}$ $\quad$ Definition of Set Union $\quad$

$\blacksquare$