Set Inequality

Theorem
$$S \ne T \iff \left({S \not \subseteq T}\right) \lor \left({T \not \subseteq S}\right)$$

Proof
$$S \ne T$$

$$\iff \lnot \left({S = T}\right)$$

$$\iff \lnot \left({\left({S \subseteq T}\right) \land \left({T \subseteq S}\right)}\right)$$ Set Equality

$$\iff \lnot \left({S \subseteq T}\right) \lor \lnot \left({T \subseteq S}\right)$$ De Morgan's Laws of Logic

$$\iff \left({S \not \subseteq T}\right) \lor \left({T \not \subseteq S}\right)$$