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It is true that you can use Löwenheim Skolem to get a countable model $M$ of ZFC assuming $Con(ZFC)$. But to use Mostowski you need additionally the well foundedness of that model, which doesn't have to be true, even though that model satisfies the axiom of regularity. The point here is that $M$ 'thinks' that it is wellfounded but from the 'outside' it is not.