Subset of Empty Set iff Empty
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Let $S$ be a set.
Let $\O$ denote the empty set.
Then $S \subseteq \O$ if and only if $S = \O$.
Suppose $x \in S$.
Then since $S \subseteq \O$, it follows that $x \in \O$.
Hence $x \notin S$.
That is, $S = \O$.