Is there any proof that the regexps are "parallel"?

Question

Theory

There are an infinite number of elements A and B - they intersect somewhere.
There is a regexp (a)', which takes into itself that part from set A, before the intersection with B.
There is a regexp (b)', which takes into itself that part, from set B, which goes to the border of the intersection with A.
How to prove that there is no such value that would be accepted by both of our regexps, located in the area of ​​intersection of values, C?

Answer 1

One of the options (if quite formally) is to build finite automata for languages, build their intersection and solve the problem of language emptiness for it.