elementary set theory - Is the intersection of every non-empty family of inductive sets equal to the intersection of every inductive set?

In Halmos Naive set theory, there is the following passage (excuse my french) in his section introducing natural numbers :

In this language the axiom of infinity simply says that there exists a successor [inductive] set A. Since the intersection of every (non-empty) family of successor sets is a successor set itself (proof?), the intersection of all the successor sets included in A is a successor set $\omega$.

I have trouble seeing why one should feel the need to state the bolded part in order to derive the existence of $\omega$.
Would he not have arrived at the same conclusion had he decided to consider directly the intersection of every inductive set included in A?

Would he not also have done so had he decided to state that the intersection of every inductive set was also an inductive set instead of invoking families of inductive sets?