In my set theory lecture notes, there is the following paragraph, after proving Axiom of Choice from Zorn's lemma:
The demonstration of this theorem is a typical example of the application of Zorn's lemma: we want to prove the existence of a set with certain properties (in our case, a choice function for a set $A$). A set like this will be a maximal element of a partially ordered set that satisfies the hypothesis of Zorn's lemma (in our case, of the set of choice functions for substs of $A$, partially ordered by strict inclusion). This way, we try to form a partially ordered set out from sets that have the properties we want the set we are looking for to have (if we want a choice function, the partially ordered set will consist of choice functions). Moreover, we will usually tend to consider the simplest partial order relations as possible (in the preceding case, like in almost any case $-$but not in all of them$-$, the strict inclusion relation).
I'm puzzled by the emphasis made in the very last line of the quote; during my whole course of set theory, we have used Zorn's lemma in the very same fashion as explained above, only considering the strict inclusion relation. However, with this method we have been able to proof a lot of results: Hausdorff 's maximal principle, Zermelo's well-ordering theorem, Teichmüller-Tukey's lemma, the order extension property, that for all infinite set $A,\;A\approx A\times A$, and many others, including the basic fact from linear algebra, that every vector space has a basis.
It is reasonable to consider, from the list I gave above, that Zorn's lemma is an extremely powerful result, because just considering the simplest partial order relation we can obtain a handful of results (although most of them are actually equivalent to Zorn's lemma).
How far-reaching can Zorn's lemma be, if we are considering non-standard partial order relations, different from the strict inclusion? Is there any concrete example of such application? I have been searching for such a result for the last days, but I have not yet encountered what I was looking for.
Thanks in advance for your interest, and your contributions