Ordinal Class is Strongly Well-Ordered by Subset

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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 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

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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.

$\blacksquare$