Assume that there exists a bijection [itex]\phi:X\to Y[/itex]. Also assume there exists some subsets [itex]A\subset X[/itex] and [itex]B\subset Y[/itex] such that a bijection [itex]\varphi:A\to B[/itex] exists too. Now Zorn's lemma implies that there exists a bijection [itex]\psi:X\to Y[/itex] such that [itex]\psi(A)=B[/itex]. I think I have now understood how to apply Zorn's lemma to things like this, so the above claim is clear to me (I assume). My question is that if [itex]A[/itex] and [itex]B[/itex] and countable, will the result also hold without Zorn's lemma?