intuitive set theory / empty set intersection question

Question

Let's say A=empty set and every element of A also a set, what is the intersection of all elements of A?

real answer:?

possible answer: universal set (I found it via using Russel paradox. And then I thought backward.)

Answer 1

Every element of the empty set is a set, no need to assume that. But the intersection of the elements of the empty set is undefined; the definition of intersections applies only to non-empty families of sets.

(Why is it undefined? If we try to apply the definition, with $I$ being the intersection of the elements of the empty set, we get $$x\in I\iff\forall y\in \emptyset\,\,\,\, x\in y.$$But that says that every $x$ is in $I$, and there is no "universal set" in modern set theory)