elementary set theory - Axiom of Choice and $mathfrak c = 2^ {aleph_0}$

I have a question about the famous Cantor equality

$$\mathfrak{c} = 2^ {\aleph_0}$$

Does the proof use Axiom of Choice at all?

It appears to me that the CBS Theorem is all that's needed (so no AC needed),

BUT AC may play a role in setting up the injections (from $\mathfrak c$ to $2^ {\aleph_0}$ and from $2^ {\aleph_0}$ to $\mathfrak c$).

related:
Requirements for Cantor Bernstein Schroder Theorem