Suppose we are working in $\mathsf{ZF-Foundation}$. Additionally, let us assume that $V = \bigcup\{V_{\alpha} | \alpha \in \mathsf{Ord}\}$ holds. Now I want to show that this implies Foundation. I am not sure if my proof works:
Let $x$ be a nonempty set, i.e. $x \in V$. Then $x \in V_\alpha$ for some $\alpha \in \mathsf{Ord}$. Now, pick $y \in x$ with $\beta$ minimal such that $y \in V_\beta$. This is possible since $\mathsf{Ord}$ is a wellfounded. Now, if $z \in x \cap y$, then $z \in y$ and hence $z \in V_\gamma$ for some $\gamma < \beta$. But since $z \in x$ as well, this contradicts the minimality of $y$. Hence, $x \cap y = \emptyset$.
Does this proof work? I am unsure, if I can use that $\mathsf{Ord}$ is wellfounded. But this shouldn't depend on Foundation, right?