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:
 * $\left\vert{\omega}\right\vert \le \omega$

Moreover, for any $n \in \omega$, by Cardinal of Finite Ordinal:
 * $\left\vert{n}\right\vert < \left\vert{n+1}\right\vert \le \left\vert{\omega}\right\vert$

Thus by Cardinal of Finite Ordinal:
 * $n \in \left\vert{\omega}\right\vert$

Therefore:
 * $\omega = \left\vert{\omega}\right\vert$

By Cardinal of Cardinal Equal to Cardinal: Corollary:
 * $\omega \in \mathcal N’$