Empty Set as Subset

Theorem
Let $S$ be a set.

Let $A$ be a subset of $S$.

Then:
 * $A = \O \iff \forall x \in S: x \notin A$

Proof
Sufficient condition follows by definition of empty set.

For necessary condition assume that:
 * $\forall x \in S: x \notin A$

Let $x$ be arbitrary.

that:
 * $x \in A$

By definition of subset:
 * $x \in S$

By assumption:
 * $x \notin A$

Thus this contradicts:
 * $x \in A$