Class of All Ordinals is Well-Ordered by Subset Relation
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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 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 1
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.
$\blacksquare$
Proof 2
From Class of All Ordinals is $g$-Tower, $\On$ is a $g$-tower.
The result follows from $g$-Tower is Well-Ordered under Subset Relation.
$\blacksquare$