Intersection is Idempotent

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Theorem

$S \cap S = S$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle S \cap S$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \in S$	$\land$	$\displaystyle x \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of set intersection
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Idempotence

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle S \cap S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\land\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of set intersection
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Idempotence