Hi

The form of the problem can suggest the use of Zorn's lemma. Now the point is to find a good set where to apply it, and I suggest you to try to find one before reading what follows!

But here may be a solution (I did not check it properly):

Consider ordered by iff and

Prove is inductive, consider a maximal element and show we must have (for instance by contradiction, assume there is an and extend into a total order of )