Empty Set is Subset of All Sets/Proof 1

Proof
By the definition of subset, $\O \subseteq S$ means:


 * $\forall x: \paren {x \in \O \implies x \in S}$

By the definition of the empty set:


 * $\forall x: \neg \paren {x \in \O}$

Thus $\varnothing \subseteq S$ is vacuously true.