The question I have is trying to show that the set $V_{\omega}$ satisfies the axiom of extensionality, and foundation, where $V_{\omega}$ is the set defined below. $$V_\omega:=\bigcup\limits_{n \in \mathbb{N}}V_n $$ where $V_n$ itself is defined as: \begin{equation*} V_n:= \begin{cases} V_0= \phi \\ V_{n+1}=V_n \cup \mathcal{P}(V_n) \\ \end{cases} \end{equation*}
$\bullet$ Extensionality Two sets are equal if and only if they have the same members.
$\bullet$ Foundation Every set $X$ has a $\epsilon-$ minimal member, that is an $x \in X$ such that $y\in x$ for no $y \in X$.
I have the following corollary's which I can use, which I have listed below:\
$\underline{\textit{Corollary 1:}}$ $\bullet$ $V_{\omega}$ satisfies the last two conditions for inaccessibility
2) If $X \in V_{\omega} $ implies $\mathcal{P}(X) \in V_{\omega}$
3) $X \in V_{\omega}$ implies that the range of any function with domain X and values in $V_{\omega}$ is a member of $V_{\omega}$.
$\underline{\textit{Corollary 2:}}$ $\forall n\in \mathbb{N},\quad V_{n+1}=\mathcal{P}(V_n)$\
$\underline {\textit{Corollary 3:}}$ If $X \in V_n$ then $X \subseteq V_n$.
I managed to get the other axioms done, but I am pretty lost on these ones. I can do the forward direction for the axiom of extensionality, but cannot get the reverse direction. Foundation I think I might have to use induction for, but i'm not sure. Help would be greatly appreciated.
If you haven't proved that yet, show that $V_\omega$ is transitive. Namely, if $x\in V_\omega$, then $x\subseteq V_\omega$.
For Extensionality, if $x\neq y$, then there is some $z\in x$ such that $z\notin y$ (or the other way around). But since $V_\omega$ is transitive, $z\in V_\omega$. What does that tell you?
To show Foundation holds, if $x\in V_\omega$, show there is a function mapping each $y\in x$ to the least $n$ such that $y\in V_n$. Now take the least $n$ for which $x\cap V_n$ is non-empty, and take $y$ in that intersection. What can you say about $y\cap x$?