User:Dfeuer/Ordinal Class is Ordinal

Theorem

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

Proof

Let $n \in \On$.

Let $m \in n$.

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

Thus $\On$ is a transitive class.

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

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

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

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

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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, $\On$ is an ordinal.

$\blacksquare$