Isomorphic Ordinals are Equal
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Theorem
Let $A$ and $B$ be ordinals that are order isomorphic.
Then $A = B$.
Proof 1
Let $S \cong T$.
Aiming for a contradiction, suppose that $S \ne T$.
Then from the corollary to Relation between Two Ordinals, either:
- $S$ is an initial segment of $T$
or:
- $T$ is an initial segment of $S$.
But as $S \cong T$, from Well-Ordered Class is not Isomorphic to Initial Segment, neither $S$ nor $T$ can be an initial segment of the other.
From this contradiction it follows that $S = T$.
$\blacksquare$
Proof 2
From Well-Ordered Class is not Isomorphic to Initial Segment, neither $A$ nor $B$ can be an initial segment of the other.
By definition, every element of an ordinal is an initial segment of it.
Hence, neither $A$ nor $B$ can be an element of the other.
By Ordinal Membership is Trichotomy, it follows that $A = B$.
$\blacksquare$