= Solution
Every <finite ordinal> belongs to $V_\omega$, so $\omega\subseteq V_\omega$ and $\omega\in\mathcal P(V_\omega)=V_{\omega+1}$. However, $\omega\notin V_\omega$, whose elements are the <hereditarily finite sets>.
Suppose $y\in V_{\omega+1}$ were an internal <power set> of $\omega$. Internally $\omega\subseteq\omega$, and $\omega$ is an available element of the structure. Thus the <Axiom of power set> would require $\omega\in y$. But $y\in\mathcal P(V_\omega)$ means $y\subseteq V_\omega$, which would put $\omega\in V_\omega$, a contradiction. Therefore
$$
\boxed{(V_{\omega+1},\in)\not\models\mathrm{PowerSet}.}
$$
This proves <power-set failure at the first rank above V omega> by an internal obstruction, including the subset $\omega$ itself.
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