Proposition 4.7.2.8 (Cantor's Diagonal Argument). Let $S$ be a set, and let $P(S)$ denote the collection of all subsets of $S$. Then $|S| < | P(S) |$.
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Proof. The construction $s \mapsto \{ s\} $ determines an injection from $S$ to $P(S)$, which shows that $|S| \leq | P(S) |$. To show that the inequality is strict, it suffices to observe that no function $f: S \rightarrow P(S)$ can be surjective, since the set $T = \{ s \in S: s \notin f(s) \} $ is an element of $P(S)$ which does not belong to the image of $f$. $\square$