Subset of Empty Set iff Empty

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Theorem

Let $S$ be a set.

Let $\O$ denote the empty set.

Then $S \subseteq \O$ if and only if $S = \O$.

Proof

Suppose $x \in S$.

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

Hence $x \notin S$.

That is, $S = \O$.

$\blacksquare$