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.

Aiming for a contradiction, suppose that:

$x \in A$

By definition of subset:

$x \in S$

By assumption:

$x \notin A$

Thus this contradicts:

$x \in A$

$\blacksquare$