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 $V \left({ x }\right)$ denote the von Neumann hierarchy on the ordinal $x$.

Then $x$ is a subset of the rank of $S$ $S \notin V \left({x}\right)$.

Necessary Condition
Let $x \subseteq \operatorname{rank} \left({ S }\right)$.

Then by Von Neumann Hierarchy Comparison:


 * $S \in V \left({x}\right) \implies S \in V \left({\operatorname{rank} \left({S}\right)}\right)$

But by Ordinal Equal to Rank:
 * $S \notin V \left({\operatorname{rank} \left({S}\right) }\right)$

By contraposition:
 * $S \notin V \left({x}\right)$

Sufficient Condition
Let $S \notin V \left({x}\right)$.

Then: