Set Equals Itself

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Theorem

All sets are equal to themselves:

$\forall S: S = S$

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle S \subseteq S \land S \supseteq S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Subset of Itself
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle S = S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Set equality

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle S \subseteq S \land S \supseteq S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Subset of Itself
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle S = S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Set equality