Subset of Empty Set iff Empty

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Theorem

Let $S$ be a set.

Let $\varnothing$ denote the empty set.


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


Proof

Suppose $x \in S$.

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

Hence $x \notin S$.

That is, $S = \varnothing$.

$\blacksquare$