This follows from consistency (and is one reason why consistency is important). If there were some whole number solution to a^n+b^n=c^n for n>2, then this would be provable in Peano Arithmetic (a much simpler system than ZFC, that axiomatizes natural number arithmetic). ZFC extends PA (every theorem of PA is a theorem of ZFC, interpreted correctly; this is relatively easy to show); thus, anything ZFC proves is not falsifiable in PA, otherwise we'd have a contradiction provable in ZFC (& it wouldn't be consistent).
Contrast this with a discipline in which there is jargon and arguments about that jargon. It might be the case that the jargon and the basic ways of arguing about the jargon forms a consistent system in some sense, even though the whole enterprise is meaningless.