# Bounded Rank implies Small Class

Jump to navigation
Jump to search

## 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

*This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.*

*If you see any proofs that link to this page, please insert this template at the top.*

*If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.*

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.

$\blacksquare$

## Sources

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