Given that sets A, B, C are subsets of a finite set S prove that

So I started of throwing a dummy element into the sets that I will keep track of:

Then I list out all the possible cases:

http://quicklatex.com/cache3/ql_c87e...40afa14_l3.png

Since in all the cases, the cardinality are equal to 1, can I say that hence the theorem is proven?