Minimal Infinite Successor Set is Infinite Cardinal

Theorem
$\omega$, the minimal infinite successor set, is an element of the infinite cardinal class $\NN'$.

Proof
By Cardinal Number Less than Ordinal: Corollary:
 * $\card \omega \le \omega$

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

Thus by Cardinal of Finite Ordinal:
 * $n \in \card \omega$

Therefore:
 * $\omega = \card \omega$

By Cardinal of Cardinal Equal to Cardinal: Corollary:
 * $\omega \in \NN'$