I have a proof of the statement that for all ordinals $\alpha, \beta$:
$$ \alpha < \beta \implies V_\alpha \in V_\beta $$
where $V_0 = \varnothing, V_{\alpha^+} = \mathcal{P}(V_\alpha),$ and $V_\lambda = \bigcup_{\gamma < \lambda} V_{\gamma}$ if $\lambda$ is a limit ordinal.
I was wondering if someone could verify whether this is a fine proof for ZFC or whether I am missing something.
I have a proof of the statement that for all ordinals $\alpha, \beta$:
$$ \alpha < \beta \implies V_\alpha \in V_\beta $$
where $V_0 = \varnothing, V_{\alpha^+} = \mathcal{P}(V_\alpha),$ and $V_\lambda = \bigcup_{\gamma < \lambda} V_{\gamma}$ if $\lambda$ is a limit ordinal.
I was wondering if someone could verify whether this is a fine proof for ZFC or whether I am missing something.
We proceed by Transfinite Induction. Suppose for the induction that for all $\gamma < \beta$ the statement holds i.e. $\alpha < \gamma \implies V_\alpha \in V_\gamma$.
Now consider $\beta$ and any $\alpha < \beta$. We wish to show $V_\alpha \in V_\beta$. To do so, we do some case distinction. Note if $\beta = 0$, then the entire statement is vacuously true.
Case 1: $\beta$ is a successor ordinal.
In this case, we know $\beta = \gamma^+$ for some $\gamma \in \textbf{ON}$. Then, by definition, we know $V_\beta = V_{\gamma^+} = \mathcal{P}(V_\gamma)$. Since $\alpha < \beta = \gamma^+ = \gamma \cup \{ \gamma \}$, we either have $\alpha = \gamma$ or $\alpha \in \gamma$. If $\alpha = \gamma$, then clearly $V_\alpha = V_\gamma$ and since evidently $V_\alpha \subseteq V_\alpha = V_\gamma$, we have $V_\alpha \in \mathcal{P}(V_\gamma)$ and thus $V_\alpha \in V_\beta$ (since $V_\beta = V_{\gamma^+} = \mathcal{P}(V_\gamma)$). Now consider the alternative where $\alpha \in \gamma$ i.e. $\alpha < \gamma$. Then, by the inductive hypothesis, we know $V_\alpha \in V_\gamma$. Moreover, since $V_\gamma$ is transitive (by part a), $V_\alpha \in V_\gamma$ implies that $V_\alpha \subseteq V_\gamma$. Now we are in a similar situation as before since $V_\alpha \subseteq V_\gamma$ implies $V_\alpha \in \mathcal{P}(V_\gamma) = V_\beta$ as required.
Case 2: $\beta$ is a limit ordinal.
In this case, we know $V_\beta = \bigcup_{\gamma < \beta} V_{\gamma}$. Since $\alpha < \beta$ and $\beta$ is a limit ordinal, we know $\alpha^+ < \beta$ as well. Then, since $\alpha^+ < \beta$, by our inductive hypothesis, the fact that $\alpha < \alpha^+$ gives us that $V_\alpha \in V_\alpha^+$ (or using the fact that $V_\alpha \subseteq V_\alpha$). Since $V_beta$ is equal to $\bigcup_{\gamma < \beta} V_{\gamma}$ and $\alpha^+ < \beta$, this lets us deduce $V_\alpha \in V_\alpha^+$ implies $V_\alpha \in V_\beta$.
In either case, we can use the inductive hypothesis to conclude that $V_\alpha \in V_\beta$. Thus by the Transfinite Induction Principle, for any $\alpha, \beta \in \textbf{ON}$, $\alpha < \beta \implies V_\alpha \in V_\beta$.