Empty Set is Subset of All Sets/Proof 1

Theorem

The empty set $\O$ is a subset of every set (including itself).

That is:

$\forall S: \O \subseteq S$

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 $\O \subseteq S$ is vacuously true.

$\blacksquare$