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.