# Class of Cardinals Contains Minimal Infinite Successor Set

## Theorem

Let $\mathcal N$ denote the class of all cardinal numbers.

Then:

$\omega \subseteq \mathcal N$

Where $\omega$ denotes the minimal infinite successor set.

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

Suppose $n \in \omega$.

By Cardinal of Finite Ordinal, $n = \left|{ n }\right|$.

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

$\blacksquare$