Union is Commutative

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Theorem

$S \cup T = T \cup S$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle \left({S \cup T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \in S$	$\lor$	$\displaystyle x \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Union
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \in T$	$\lor$	$\displaystyle x \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Commutation
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle \left({T \cup S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Union

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle \left({S \cup T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\lor\)	\(\displaystyle x \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Union
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in T\)	\(\lor\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Commutation
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle \left({T \cup S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Union