Set has Rank/Proof 1

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

Suppose that all the elements $a \in S$ have a rank.

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

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

be the rank of $a$.

Let:
 * $\ds y = \sup \set {\map F a : a \in S}$

be the least level of the Von Neumann Hierarchy containing all elements of $S$.

Then, for any $a \in S$:

Therefore $S \in \map V z$ for some ordinal $z = y + 1$.

Thus by Epsilon Induction every set has a rank.