Empty Set as Subset

Theorem
Let $S$ be a set.

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

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

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.

Aiming for a contradiction suppose that
 * $x \in A$

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

By assumption:
 * $x \notin A$

Thus this contradicts
 * $x \in A$