No Injection from Power Set to Set/Lemma

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\mathcal P \left({S}\right)$.

Proof
Suppose for the sake of contradiction that there is such a $B$.

Let $i: B \to S$ be an injection.

Let $f: B \to \mathcal P \left({S}\right)$ be a surjection.

Let $i^\gets: \mathcal P \left({S}\right) \to \mathcal P \left({B}\right)$ be the mapping induced on $\mathcal P \left({S}\right)$ by the inverse $i^{-1}$.

By Mapping Induced by Inverse of Injection is Surjection, $i^\gets$ is a surjection.

Let $f^\to: \mathcal P \left({B}\right) \to \mathcal P \left({ \mathcal P \left({S}\right)}\right)$ be the mapping induced by $f$.

By Mapping Induced on Power Set by Surjection is Surjection, $f^\to$ is a surjection:

We have that $i^\gets: \mathcal P \left({S}\right) \to \mathcal P \left({B}\right)$ and $f^\to: \mathcal P \left({B}\right) \to \mathcal P \left({ \mathcal P \left({S}\right) }\right)$ are surjective.

By Composite of Surjections is Surjection, their composition $f^\to \circ i^\gets: \mathcal P \left({S}\right) \to \mathcal P \left({\mathcal P \left({S}\right)}\right)$ is a surjective by Composite of Surjections is Surjection.

But this violates Cantor's Theorem, contradicting the assumption that such a $B$ exists.