# 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

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Let $V$ denote the von Neumann hierarchy.

Then:

 $\displaystyle \forall x \in S: \ \$ $\displaystyle \operatorname{rank} \left({x}\right)$ $\le$ $\displaystyle y$ by hypothesis $\displaystyle \implies \ \$ $\displaystyle x$ $\in$ $\displaystyle V \left({y + 1}\right)$ Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy and Ordinal Equal to Rank $\displaystyle \implies \ \$ $\displaystyle S$ $\subseteq$ $\displaystyle V \left({y + 1}\right)$ Definition of Subset

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

$\blacksquare$