No Injection from Power Set to Set/Proof 1

Theorem
Let $S$ be a set.

Let $\mathcal P(S)$ be the power set of $S$.

Then there is no injection from $\mathcal P(S)$ into $S$.

Proof
Suppose for the sake of contradiction that $f: \mathcal P(S) \to S$ is an injection.

By Injection has Surjective Left Inverse Mapping, there is a surjection $g: S \to \mathcal P(S)$.

But this contradicts Cantor's Theorem.

Thus there can be no such injection.