A reference for undecidability of "$|A|> |B|$ implies $|\mathcal{P}(A)|>|\mathcal{P}(B)|$" in ${\rm ZFC}$.

Question

Note: This is a reference-request question and thus doesn't need the usual type of context.

According to this:

It is a surprising fact that the statement $$A > B \implies \mathcal{P}(A) > \mathcal{P}(B)$$ is undecidable in $\mathrm{ZFC}$.

. . . and . . .

we define $A > B$ to mean: there exists an injection $B\to A$ but no bijection between these two sets.

Please may I have a reference for this?

Context:

It is indeed surprising. I can't say that I would understand a proof of it, but having a reference would go a long way to convincing whoever that it's true; it's the kind of thing I would share with fellow students. If the reference includes some prerequisites for the proof, that would be ideal.

Answer 1

Cohen's original work would be enough.

Cohen, Paul Joseph, The independence of the continuum hypothesis. I, II, Proc. Natl. Acad. Sci. USA 50, 1143-1148 (1963); 51, 105-110 (1964). ZBL0192.04401.

The second paper, Lemma 21 and Lemma 22 give us exactly that if we are adding $\kappa$ many Cohen reals, and $\kappa^{\omega_1}=\kappa$ in the ground model, then $2^{\aleph_1}=\kappa$ in the extension, and therefore $2^{\aleph_0}=2^{\aleph_1}$ in that case.