Assuming it contained only sets that could exist within a model of ZF, a set of finite sets that contain only finite sets must be countable, correct?

My reasoning, assuming I understand correctly, is that it follows from $L_{\omega} = V_{\omega}$ where $L$ and $V$ are as generally described here: https://en.m.wikipedia.org/wiki/Constructible_universe

Would $\{x \in L_{\omega} : |x| < |\mathbb{N}| \}$ in fact equal the set of finite sets that contain only finite sets?

My reasoning, assuming I understand correctly, is that it follows from $L_{\omega} = V_{\omega}$ where $L$ and $V$ are as generally described here: https://en.m.wikipedia.org/wiki/Constructible_universe

Would $\{x \in L_{\omega} : |x| < |\mathbb{N}| \}$ in fact equal the set of finite sets that contain only finite sets?

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