NP-complete is the intersection of sets NP and NP-hard , but not the part of NP that contains P.
Lots of NP-hard problems are not in NP is important.
https://en.wikipedia.org/wiki/NP-hardness#/media/File:P_np_n...
https://en.wikipedia.org/wiki/NP-hardness#/media/File:P_np_n...