# Class of All Ordinals is Well-Ordered by Subset Relation

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## Theorem

Let $\On$ be the class of all ordinals.

Then the restriction of the subset relation, $\subseteq$, to $\On$ is a well-ordering.

That is:

$\subseteq$ is an ordering on $\On$.
If $A$ is a non-empty subclass of $\On$, then $A$ has a smallest element under the subset relation.

## Proof 1

By Subset Relation on Class is Ordering, $\subseteq$ is an ordering of any class.

Let $A$ be a subclass of $\On$.

By Intersection of Ordinals is Smallest, $A$ has a smallest element under the subset relation.

$\blacksquare$

## Proof 2

From Class of All Ordinals is $g$-Tower, $\On$ is a $g$-tower.

The result follows from $g$-Tower is Well-Ordered under Subset Relation.

$\blacksquare$