You can't prove such a thing in ZFC [1]. ZFC's "power set axiom" is a misnomer. It doesn't imply the existence of infinite power sets. It just says if any set S exists and is a subset of an infinite set A, then S is element of a set P (the supposed "power set"). But the axiom doesn't imply the existence of "all" subsets of A. There is no way for first-order logic to state "every possible combination of elements of A forms a set". And Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC. [2]
[1] Or any other theory of first-order logic.
[2] This is explained in more detail in Stewart Shapiro's book "Foundation without Foundationalism" p. 114f.
[1] Or any other theory of first-order logic.
[2] This is explained in more detail in Stewart Shapiro's book "Foundation without Foundationalism" p. 114f.