Exists Element Not in Set/Proof 1

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.