No Injection from Power Set to Set/Proof 1

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

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

But this contradicts Cantor's Theorem.

Thus there can be no such injection.