For a set X I will write X < Y to indicate that the cardinality of X is strictly smaller than the cardinality of Y. Let me write P(X) for the power set of X. Consider the proposition:
Prop: If P(X) < P(Y)then X < Y.
Is it possible to prove this in ZFC (without continuum hypothesis)? If not is it perhaps the case that the Prop is equivalent to generalized continuum hypothesis? It is trivial to show that generalized continuum hypothesis implies the proposition, but what about the other direction?