Minimal Infinite Successor 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:

$\omega \in \mathcal N’$

$\blacksquare$