Exists Element Not in Set

Theorem

Let $S$ be a set.

Then $\exists x: x \notin S$.

That is, for any set, there exists some element which is not in that set.

Proof

Consider the power set $\mathcal P \left({S}\right)$ of $S$.

Suppose $\forall x \in \mathcal P \left({S}\right): x \in S$.

Then the identity mapping $I_S :S \to \mathcal P \left({S}\right)$ would be a surjection.

But from Cantor's Theorem, there is no surjection $f: S \to \mathcal P \left({S}\right)$.

Thus $\exists x \in \mathcal P \left({S}\right): x \notin S$.

Hence the result.

$\blacksquare$