Bounded Rank implies Small Class

Theorem
Let $S$ be a class.

Suppose the rank, denoted $\map {\operatorname{rank} } x$, of each $x \in S$ is bounded above by some ordinal $y$.

Then $S$ is a small class.

Proof
Let $V$ denote the von Neumann hierarchy.

Then:

Therefore, by Axiom of Subsets Equivalents, it follows that $S$ is a small class.