Ordinal Class is Strongly Well-Ordered by Subset

Theorem

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

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

That is:

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

Proof

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

By Subset Relation is Ordering, $\subseteq$ is an ordering of any set.
Let $A$ be a subclass of $\On$.
$\blacksquare$