Set Inequality

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Theorem

$S \ne T \iff \paren {S \nsubseteq T} \lor \paren {T \nsubseteq S}$


Proof

\(\ds S \ne T\) \(\iff\) \(\ds \neg \paren {S = T}\)
\(\ds \) \(\iff\) \(\ds \neg \paren {\paren {S \subseteq T} \land \paren {T \subseteq S} }\) Definition 2 of Set Equality
\(\ds \) \(\iff\) \(\ds \neg \paren {S \subseteq T} \lor \neg \paren {T \subseteq S}\) De Morgan's Laws: Disjunction of Negations
\(\ds \) \(\iff\) \(\ds \paren {S \nsubseteq T} \lor \paren {T \nsubseteq S}\)

$\blacksquare$