Minimally Inductive Set is Infinite Cardinal

Theorem
$\omega$, the minimal infinite successor set, is an element of the infinite cardinal class $\mathcal N’$.

Proof
By Cardinal Number Less than Ordinal/Corollary, $\left|{ \omega }\right| \le \omega$.

Moreover, for any $n \in \omega$, it follows that $|n| < |n+1| \le |\omega|$ by Cardinal of Natural Number.

It follows that $n \in |\omega|$ by Cardinal of Natural Number.

Therefore, $\omega = \left|{ \omega }\right|$.

By Cardinal of Cardinal Equal to Cardinal/Corollary, $\omega \in \mathcal N’$.