Every Set in Von Neumann Universe

Theorem
Let $S$ be a small class.

Then $S$ is well-founded.

Proof
The proof shall proceed by Epsilon Induction on $S$.

Suppose that all the elements $a \in S$ are well-founded.

That is, $a \in \map V x$ for some $x$.

Let:
 * $\ds \map F a = \bigcap \set {x \in \On : a \in \map V x}$

Take $\ds \bigcup_{a \mathop \in S} \map F a$.

Take any $a \in S$.

Therefore:
 * $\ds S \in \map V {\bigcup_{x \mathop \in S} \map F x + 1}$

and $S \in \map V x$ for some ordinal $x$.