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 $\powerset S$ of $S$.

$\forall x \in \powerset S: x \in S$.

Then the identity mapping $I_S: S \to \powerset S$ would be a surjection.

But from Cantor's Theorem, there is no surjection $f: S \to \powerset S$.

From this contradiction it follows that:


 * $\exists x \in \powerset S: x \notin S$

Hence the result.