Subset of Empty Set iff Empty

Theorem
Let $S$ be a set.

Let $\varnothing$ denote the empty set.

Then $S \subseteq \varnothing$ $S = \varnothing$.

Proof
Suppose $x \in S$.

Then since $S \subseteq \varnothing$, it follows that $x \in \varnothing$.

Hence $x \notin S$.

That is, $S = \varnothing$.