# Class of Cardinals Contains Minimal Infinite Successor Set

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## 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

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Suppose $n \in \omega$.

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

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

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.39$