Hr e says about the axiom of power set: What 'goes wrong' is the rest of the axiom: y does not contain every subset of x; it only contains those subsets of x that are in M. So it is perfectly possible that this 'powerset' of x is countable. [Tim Chow: "Forcing for dummies", sci.math.research (10 Mar 2001)]
Of course here he needs to address dummies. Everyone else would see through clearly: If a set x is in the model M, but a subset of x is missing in the model M, then at least one element of this x must be missing in the model M, then x is not in the model M
In particular, If |N is in the model M, but a subset of |N is missing in the model M, then at least one element of |N must be missing in the model M, then |N is not in the model M, and M is not even a model of ZF for violating the axiom of infinity.
Therefore the model M is uncountable from outside. A countable model of ZFC and a model of whole ZFC are mutually incompatible. Skolem's proof implies that ZFC cannot have any model.