No Injection from Power Set to Set

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
The identity mapping $f: \mathcal P(S) \to \mathcal P(S)$ is a surjection by Identity Mapping is Surjection.

Thus by the lemma, there can be no injection from $\mathcal P(S)$ into $S$.