# Class of All Ordinals is Ordinal

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## Theorem

The class of all ordinals $\On$ is an ordinal.

## Proof

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The epsilon relation is equivalent to the strict subset relation when restricted to ordinals by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

It follows that:

- $\forall x \in \On: x \subset \On$

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The initial segment of the class of all ordinals is:

- $\set {x \in \On : x \subset \On}$

This article, or a section of it, needs explaining.In particular: What is meant by $\subset$ here: $\subseteq$ or $\subsetneqq$? Sorry, but "makes no sense" is a bit strong: "initial segment" has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ in the context of a well-ordered set, but the term makes perfect sense if defined in the context of a class. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Therefore, by the definition of ordinal, $\On$ is an ordinal.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $7.12$