Löwenheim-Skolem and proper class models of ZFC.

Question

Let $N$ be a proper class model of ZFC and $x \subset N$ a set. Show that there is a set $y \in N$ such that $x \subset y$.

If $x \subset N$, I think that by the downward's part of Löwenheim-Skolem, we can find an elementary submodel $y$ of $N$ such that $x \subset y$. But how would I conclude that $y \in N$? It seems like I'm missing something trivial but I can't figure out what exactly.

Note: A hint to point me to the right direction would be more appreciated than a complete answer.

Answer 1

Without loss of generality $x$ is transitive, at least relative to $N$ (that is, $\bigcup x\cap N\subseteq x$).

Because $N$ is a proper class model, it contains elements of arbitrarily large cardinalities. In particular it contains a cardinal number $\kappa$ that does not inject into $x$ at the metalevel, and therefore cannot inject into any subclass of $x$ in $N$.

But then $N$'s $H_\kappa$ must contain all of $x$.

Answer 2

This is a slight twist on Henning's answer:

Fix, working in $V$, an ordinal $\alpha$ such that $x \subseteq V_{\alpha}$. By the absoluteness of $\mathrm{rank}_{\in}$, we have $V^N_\alpha = V_\alpha \cap N$ and hence $$y = x \cap N \subseteq V_{\alpha} \cap N = V^N_\alpha.$$