Existence of Minimal Uncountable Well-Ordered Set/Proof Without Using Choice

Theorem
There exists a minimal uncountable well-ordered set.

That is, there exists an uncountable well-ordered set $\Omega$ with the property that every initial segment in $\Omega$ is countable.

Proof without using choice
By the axiom of powers, there exists the power set $\mathcal P \left({\N}\right)$.

By Power Set of Natural Numbers Not Countable, this set is uncountable.

We construct a well-ordering $\left({P \left({\N}\right), \preccurlyeq}\right)$ that has the desired defining properties of $\Omega$.