Power-set failure at the first rank above V omega ($(V_{\omega +1},\in )/\models \mathrm{PowerSet}$)

Every finite ordinal belongs to $V_{\omega }$, so $\omega \in V_{\omega +1}$ but $\omega \notin V_{\omega }$. An internal power set $y$ of $\omega$ would contain $\omega$ itself. Yet $y\in V_{\omega +1}$ implies $y\subseteq V_{\omega }$, making that impossible. This example illustrates why one must check which subsets the structure sees when proving an axiom fails.