User:Dfeuer/Ordinal Class is Ordinal

Theorem
The Ordinal Class $\operatorname{On}$ is an ordinal.

Proof
Let $n \in \operatorname{On}$.

Let $m \in n$.

By Element of Ordinal is Ordinal, $m \in \operatorname{On}$.

Thus $\operatorname{On}$ is a transitive class.

Let $S$ be any non-empty subclass of $\operatorname {On}$.

Let $p$ be an arbitrary element of $S$.

If $p \cap S = \varnothing$ then $p$ is the smallest element of $S$.

Otherwise, $p \cap S$ is a non-empty subset of $p$.

Since $p$ is an ordinal, $p \cap S$ has a smallest element, which will then be the smallest element of $S$.

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