Let be the union of distinct intervals in i.e. none of the intervals are the same

I am trying to show that is a step function are disjoint. I am a bit stumped.

So assume .

If there is only one non-empty intersection of two intervals, it is easy to show that this is impossible, but I am unable to prove it in the general case, i.e. when there is an arbitrary number of collections of intervals with non-empty intersection, has anyone got any tips? Thanks very much