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 $S$ iff $S \notin V \left({x}\right)$.

Necessary Condition
If $x \subseteq \operatorname{rank} \left({ S }\right)$ then it follows


 * $S \in V\left({ x }\right) \implies S \in V\left({ \operatorname{rank} \left({ S }\right) }\right)$ by Von Neumann Hierarchy Comparison

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

By contraposition, $S \notin V\left({ x }\right)$.

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