Minimal Uncountable Well-Ordered Set Unique up to Isomorphism

Theorem
Let $\Omega, \Omega'$ be minimal uncountable well-ordered sets.

Then $\Omega$ is order isomorphic to $\Omega'$.

That is, the minimal uncountable well-ordered sets] is unique up to order isomorphism.

Proof
From Wosets are Isomorphic to Each Other or Initial Segments, precisely one of the following holds:


 * $\Omega$ is order isomorphic to $\Omega'$

or:


 * $\Omega$ is order isomorphic to an initial segment in $\Omega'$

or:


 * $\Omega'$ is order isomorphic to an initial segment in $\Omega$.

By the definition of minimal uncountable well-ordered sets, $\Omega$ and $\Omega'$ are uncountable.

The initial segments of $\Omega$ and $\Omega'$ are countable.

An uncountable set can't be isomorphic to a countable set.

Thus $\Omega$ must be order isomorphic to $\Omega'$.

The result follows from the definition of uniqueness.