I have problems understanding this axiom.

It is stated:

For any set S, there exist a set B such that $\displaystyle x \in B \ iff \ x \in A \ for \ some \ A \in S.$

The wikipedia explanation doesn't really help either:

What the axiom is really saying is that, given a set S, we can find a set B whose members are precisely the members of the members of S. By the axiom of extensionality this set B is unique and it is called the union of S, and denoted ∪S.

My (infantile) understanding:

1. There is a large set S.

2. There is also a (somewhat smaller, but not necessary) set A.

3. Some elements of A also belong to S. (corollary: some elements of A lie outside of S). (is member = element?)

4. A unique set B exist in the intersection of A and S.

This is INTERSECTION! Not union! Something about my understanding is wrong. Can anyone please help? Thanks.