Need clarification regarding certificates of coNP problems
The proof does bot need to be a subset. It might be another indicator that the given set has a some stricture preventing it from being a positive instance of the subsetsums problem. A good example with a non-trivial certificate is linear programming. Linear programs admit both a positive and a negative certificate (For the question whether the optimal can be smaller/greater than a value k). The positive instance is of course an assignment of the variable. The negative however is given by Faraks lemma and the weak duality.

A good exercise for you is to look up linear programs, weak duality and Farkas lemma :)