Class of All Ordinals is Well-Ordered by Subset Relation

Theorem
Let $\operatorname{On}$ be the class of all ordinals.

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

That is:


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

Proof
By Subset Relation is Ordering, $\subseteq$ is an ordering of any set.

Let $A$ be a subclass of $\operatorname{On}$.

By Intersection of Ordinals is Smallest, $A$ has a $\subseteq$-smallest element.