A yes answer to an NP problem must be deterministically verifiable in polynomial time. The complement is that the no answer must be similarly verifiable. If the problem is NP-complete, there will generally be an intractable number of possibilities. How can it be possible to deterministically verify the no answer for all possibilities in polynomial time?
As an example, would it not be necessary to address every non-empty subset to verify a no answer for the subset sum problem?