Minimal Infinite Successor Set is Minimal

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 $\omega \subseteq B$.