# Minimal Infinite Successor Set is Minimal

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## Theorem

The minimal infinite successor set $\omega$ is a subset of every infinite successor set.

## Proof

Let $A$ be an infinite successor set.

Let $B$ be another arbitrary infinite successor set.

Then from Intersection is Subset, $A \cap B \subseteq A$.

From Intersection of Infinite Successor Sets, $A \cap B$ is also an infinite successor set.

This set $A \cap B$ is one of the subsets

By the definition of $\omega$ it follows that $\omega \subseteq A \cap B$, and consequently that $\omega \subseteq B$.

$\blacksquare$

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 11$: Numbers