Union is finite implies the collection is finite

Question

Let $C$ be a collection of sets, and $\bigcup C \in V_{\omega}$ where $V_{\omega}$ is the collection of hereditarily finite sets. Is it possible to show that $C\in V_{\omega}$?

YES. Because $V_\omega$ models every axiom of set theory except the infinity axiom;

i.e. $\bigcup C \in V_{\omega}\:\Rightarrow\:C \subseteq \mathcal{P}\left(\bigcup C\right)\in V_{\omega}\:\Rightarrow\: C\in \mathcal{P}\left(\mathcal{P}\left(\bigcup C\right)\right)\in V_{\omega}\:\Rightarrow\: C\in V_{\omega}$.

QUESTION What if I instead assume $\bigcup C$ is simply finite. Can I show that $C$ is finite?

Initial Thoughts From the assumption, it follows that every element of $C$ is finite. Is it possible to have an infinite collection of finite sets whose union is finite?

EDIT: $C\subseteq \underbrace{\mathcal{P}(\bigcup C)}_{\text{finite}}$. "Thank you" to a recently deleted answer which brought my senses back.

Answer 1

Sorry, I accidentally deleted my response.

It's not possible for the union to be finite and the collection infinitie. The point here is that, we can note that if $U=\cup C$ then $C\subseteq P(U)$. The power set of S is finite if and only if S is finite..