# 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:

\(\displaystyle \forall x \in S: \ \ \) | \(\displaystyle \operatorname{rank} \left({x}\right)\) | \(\le\) | \(\displaystyle y\) | $\quad$ by hypothesis | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle x\) | \(\in\) | \(\displaystyle V \left({y + 1}\right)\) | $\quad$ Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy and Ordinal Equal to Rank | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle S\) | \(\subseteq\) | \(\displaystyle V \left({y + 1}\right)\) | $\quad$ Definition of Subset | $\quad$ |

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

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 9.19$