# Elementary submodels and power sets

Suppose $$M$$ and $$N$$ are transitive models of, say, Kripke-Platek set theory such that $$M \in N$$ and $$M \prec N$$. Then, for any $$x \in M$$, the set $$\{y \in M | y \subseteq x\} \in N$$ by $$\Delta_0$$-separation in $$N$$. It appears that this set would be the power set of $$x$$ and thus witness that $$n \vDash \text{"the power set of x exists"}$$. By elementarity, this would also hold in $$M$$. Since this would hold for every $$x \in M$$, that would mean that the powerset axiom holds in $$M$$, and thus in $$N$$ by elementarity. Since ZFC proves that $$H_{\omega_1}$$ is a set and (by the Löwenheim-Skolem theorem) that countable elementary submodels of this set exist, that would mean that ZFC proves that $$H_{\omega_1}$$ is a transitive model of ZFC. Where does this argument go wrong?

While writing this question, I realized that the flaw of this proof is the assumption that the set $$\{y \in M | y \subseteq x\} \in N$$ is the powerset of $$x$$ in $$N$$. Actually, it doesn't rule out that $$x$$ has more subsets in $$N$$ than in $$M$$, so that's how it is unless the powerset axiom holds in $$M$$ and $$N$$.