Kuratowski-Zorn Lemma with pre-order (quasi-order, proset) instead poset? What would happen if we would use pre-order (there is no weak-antisymmetry, only reflexivity and transitivity) in  Kuratowski-Zorn Lemma instead of partial order? 

Suppose that we have set P which is preordered and  has the property
  that every chain (i.e. totally ordered subset) has an upper bound in
  P. Then the set P contains at least one maximal element.

Is this statement true in ZFC? If not, what can we postulate to make it true? I think it can be false, or at least we need to postulate something more (or not?), but I am just missing something and don't see it.
P.S.

 I wanted to put tags with combination of words like: Kuratowski Zorn Lemma Pre-order Proset, but it seems that none of them exist and I need to have 300+ reputation to create new ones, thus I only typed (set-theory) :( If someone with bigger reputation could make this topics' tags more precise, I would be grateful :)

 A: In $\sf ZFC$ you can do the following trick.
Let $(P,\leq)$ be a preordered set. Define $x\sim y\iff x\leq y\land y\leq x$, clearly an equivalence relation. Now using the axiom of choice choose a representative from each equivalence class. This defines $P'\subseteq P$ which is a partially ordered set.
If $(P,\leq)$ satisfied the Zorn lemma assumptions, so must $(P',\leq)$. Therefore $P'$ has a maximal element, so $P$ has one as well. Where maximal in a preordered set means that there is no strictly larger element.

(As Hagen von Eitzen points out, one can looks at the partial order defined naturally on $P/\sim$, and see that it satisfies Zorn's lemma assumptions, therefore it has a maximal element, which has an equivalence class of maximal elements in $P'$.
We do not resort to using the axiom of choice directly, since we are not choosing from the representatives of each class, but since Zorn's lemma is equivalent to the axiom of choice to begin with, this does not exempt us from using the axiom of choice in its full generality anyway.)
