# Set Inequality

$S \ne T \iff \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)$
 $\displaystyle S \ne T$ $\iff$ $\displaystyle \neg \left({S = T}\right)$ $\displaystyle$ $\iff$ $\displaystyle \neg \left({\left({S \subseteq T}\right) \land \left({T \subseteq S}\right)}\right)$ 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 $\displaystyle$ $\iff$ $\displaystyle \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)$
$\blacksquare$