Bounded Rank implies Small Class

Theorem
Let $S$ be a class.

Suppose the rank, denoted $\operatorname{rank} \left({ x }\right)$, 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:

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