The steps to show something is NP-complete is to show it is a member of NP, and then show an NP-complete problem reduces to it in polynomial-time. Why is the first step necessary? Is it possible to reduce an NP-hard problem to an NP-complete problem in polynomial time? Because that would be the only instance in which this is important. 
Why do we prove X is in NP to prove it is NP-complete?
algorithms;time complexity;np complete;np