Set Inequality

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Theorem

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


Proof

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

$\blacksquare$