Exists Subset which is not Element/Proof 2

Proof
Consider the power set $\powerset S$ of $S$.

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

That is, there are more subsets of $S$ than there are elements of $S$.

So there must be at least one subset of $S$ which is not an element of $S$.