Axiom of Unions and its use of the existential quantifier

Question

I'm reading Halmos's Naive Set Theory, and right now I'm on the section about the axiom of unions. As stated in the book, the axiom reads:

For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.

Essentially, U={x: x∈X for some X in C}. My question concerns the use of the quantifier "for some". My knowledge of quantifiers and formal logic is minimal, so I'm wondering if someone can explain to me intuitively why "for some" is used instead of "for all". For when I find the union of a set, aren't I including the elements from all the sets inside the set?

Answer 1

Take the example of $A\cup B$ (or $C=\{A,B\}$ in this case). Then $x\in A\cup B$ if and only if $x\in A$ or $x\in B$ if and only if $\exists X\in\{A,B\}$ such that $x\in X$.

If you would have written "for all $X\in C$" you would get the intersection of all the sets in $C$, which can be everything if $C$ is empty, and that's not a set.