Set Inequality

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Theorem

$S \ne T \iff \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S \ne T$	$\iff$	$\displaystyle \neg \left({S = T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \neg \left({\left({S \subseteq T}\right) \land \left({T \subseteq S}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Equality of Sets
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \neg \left({S \subseteq T}\right) \lor \neg \left({T \subseteq S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	De Morgan's Laws
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \ne T\)	\(\iff\)	\(\displaystyle \neg \left({S = T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \neg \left({\left({S \subseteq T}\right) \land \left({T \subseteq S}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Equality of Sets
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \neg \left({S \subseteq T}\right) \lor \neg \left({T \subseteq S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)