I'm seeking a finite model of union, and in particular I would like to show that $V_n \vDash$ union. As usual, I have defined $V_0 = \emptyset$ and $V_n = {\cal P}(V_{n-1})$ for $n \in \omega$. Have I made a mistake in this proof?
Proof. Pick $x \in V_n$. Show $\bigcup x \in V_n$, or rather $\bigcup x \subseteq V_{n-1}$.
Since $x \in V_n$ this means that $x \subseteq V_{n-1}$. Therefore $\bigcup x \subseteq \bigcup V_{n-1} = V_{n-2} \subseteq V_{n-1}$.