Your definition of "mutual independence" is indeed correct. As you can see, it involvesintersections. So you must reduce to an expression using intersections. The usual trick is to use the complement since (where if is the whole probability space).

So here are the first steps (that you should justify): .

You will need to know that if are mutually independent, then are mutually independent as well. Either you know it already or you'll have to prove it.

Laurent.