I have this prob and still haven't figured about
Let (A, $\leq$) be a partially ordered set. A subset B of A is called an antichain if and only if for any two distinct elements x and y in B, neither x $\leq$ y or y $\leq$ x. Prove that every antichain is contained in an antichain which is maximal with respect to inclusion $\subseteq$
I tried to make a set of subsets of A which are antichains. However I don't know what to do next. The text book says use Hausdorff maximal principle, but I can't use it from the set I made cause elements are not totally ordered sets. What should I do?