Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy

Theorem
Let $x$ be an ordinal.

Let $S$ be a small class.

Let $\map V x$ denote the von Neumann hierarchy on the ordinal $x$.

Then $x$ is a subset of the rank of $S$ $S \notin \map V x$.

Necessary Condition
Let $x \subseteq \map {\operatorname{rank} } S$.

Then by Von Neumann Hierarchy Comparison:


 * $S \in \map V x \implies S \in \map V {\map {\operatorname{rank} } S}$

But by Ordinal Equal to Rank:
 * $S \notin \map V {\map {\operatorname{rank} } S}$

By contraposition:
 * $S \notin \map V x$

Sufficient Condition
Let $S \notin \map V x$.

Then: