Class of All Ordinals is Well-Ordered by Subset Relation/Proof 1

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

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.

$\blacksquare$