Proof about parity of a set of four integers

Statement to prove:

Let S={a,b,c,d} be a set of four distinct integers. Prove that if either (1) for each xES, the integer x and the sum of any two of the remaining three integers of S are of the same parity or (2) for each xES, the integer x and the sum of any two of the remaining three integers of S are of opposite parity, then every pair of integers of S are of the same parity.

I have two proofs of this statement and I want to make sure they are correct. In the first one I assume that some pair of integers of S are of different parity, and in four different cases show that neither condition (1) or condition (2) is satisfied. So proof by contrapositive.

In the second I assume that every pair of integers in S are of opposite parity. Since this is always false (right?) the contrapositive is a true statement and I'm done. So a vacuous proof of the contrapositive. I'm unsure of this one because so far I haven't seen a two proof methods combined.