Class of All Ordinals is Well-Ordered by Subset Relation

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

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

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