Minimal Infinite Successor Set is Minimal
Let $A$ be an infinite successor set.
Let $B$ be another arbitrary infinite successor set.
Then from Intersection is Subset, $A \cap B \subseteq A$.
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 $\omega \subseteq B$.