Minimal Infinite Successor Set is Ordinal/Proof 1

Proof
The minimal infinite successor set is a set of ordinals by definition.

From the corollary of ordinals are well-ordered, it is seen that $\left({\omega, \Epsilon \! \restriction_\omega}\right)$ is a strictly well-ordered set.

It is to be shown by induction on minimal infinite successor set that $\forall n \in \omega: \omega_n = n$

Induction Hypothesis
Suppose that $\omega_n = n$ for some $n \in \omega$.

Induction Step
And so $\omega$ is an ordinal.