Isomorphic Ordinals are Equal/Proof 2

Theorem

Let $A$ and $B$ be ordinals that are order isomorphic.

Then $A = B$.

Proof

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 $A = B$.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.38$