# Are there any “complete” languages in $coNP -NP$?

Suppose $$coNP \neq NP$$

language B would be called "complete" in $$coNP-NP$$ if:

1. $$B\in coNP - NP$$
2. $$A\in coNP-NP \implies A\leq_pB$$

Are there any "complete" languages in $$coNP - NP$$?

If we are assuming that $$coNP≠𝑁𝑃$$,
we can conclude that every language that is $$co NP$$ complete is not in $$NP$$ (a contradiction to your given assumption).
Thus, every language we already know of that is $$coNP$$ complete, is complete as well in $$coNP -NP$$.