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

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


$B\in coNP - NP$
$A\in coNP-NP \implies  A\leq_pB$


Are there any "complete" languages in $coNP - NP$?
If we are assuming that $coNP≠$,

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 $coNP -NP$.