# Bounded Rank implies Small Class

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## 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:

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

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

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

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*: $\S 9.19$