This cannot be proved from the axioms of $\sf ZFC$ (and so certainly not from naive set theory) but it cannot be refuted either.
That is to say, assuming that the axioms of set theory (read: $\sf ZFC$) are consistent there are models of set theory in which $2^X\sim 2^Y\implies X\sim Y$, and there are other models in which there are $X\nsim Y$ such that $2^X\sim 2^Y$.
For example if $\sf GCH$ holds then the statement is true, because the power set is "as small as possible", but it is consistent that there is an uncountable set of real numbers whose power set is equipotent with the real numbers themselves, i.e. $X$ such that $\Bbb N<X$ but $2^X\sim 2^\Bbb N$.
The statement is weaker than $\sf GCH$, and in a related (but unrelated) post on MathOverflow I called it "Injective Continuum Function", ICF. I have seen mentioning that this was called by Tarski "Weak Power Hypothesis", WPH.