Intersection is Commutative

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Theorem

$S \cap T = T \cap S$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle \left({S \cap T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \in S$	$\land$	$\displaystyle x \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Intersection
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \in T$	$\land$	$\displaystyle x \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Commutation
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle \left({T \cap S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Intersection

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle \left({S \cap T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\land\)	\(\displaystyle x \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Intersection
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in T\)	\(\land\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Commutation
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle \left({T \cap S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Intersection