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 | |||||||||||
\(\ds \) | \(\iff\) | \(\ds \paren {S \nsubseteq T} \lor \paren {T \nsubseteq S}\) |
$\blacksquare$