The transitive collapse of an ultraproduct of $V$ is an inner model

Question

So this is the situation: Let $U$ be an ultrafilter over a set $S$ and let $M_U$ be the transitive collapse of $\text{Ult}(V, U)$. $M_U$ is an inner model of ZFC.

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Thus far i have used Los's theorem to prove that for every sentence $\varphi$, $\text{Ult}(V, U) \models \varphi \leftrightarrow \{i \in S: V \models \varphi \} \in U$, which was easy and shows that $M_U$ is a model of ZFC. But now i have no idea on how to show that $\text{Ord} \subseteq M_U$. I would be really grateful for any hints or answers.

Answer 1

Either show that the embedding from $V$ to $M_U$ does not lower ranks.

Or else simply note that $M_U$ is unbounded in size (as the embedding is injective) and for any $x \in M_U$, $\operatorname{rk}(x) \in M_U$ since $M_U$ satisfies ZFC (and $\operatorname{rk}$ is absolute for transitive models).