I would use induction to make matters simple. This way you only need to do it for two disjoint sets.
Let A and B be disjoint finite sets. Then .
Think you can take it from there?
Yeah sure. So we proved about the base case when you adjoin 1 disjoint set to A that is how you count them.
So suppose for induction hypothesis that for disjoint sets we have , we now show it must be true for adjoining n+1 disjoint sets. So are all disjoint. By induction hypothesis . Now we notice so by our base case in the first post
Which completes the induction.