Editing Zorn's lemma (section)

== Proof ==
The basic idea of the proof is to reduce the proof to proving the following weak form of Zorn's lemma:

{{math_theorem|name=Lemma|math_statement=Let <math>F</math> be a set consisting of subsets of some fixed set such that <math>F</math> satisfies the following properties:
# <math>F</math> is nonempty.
# The union of each totally ordered subsets of <math>F</math> is in <math>F</math>, where the ordering is with respect to set inclusion.
# For each set <math>S</math> in <math>F</math>, each subset of <math>S</math> is in <math>F</math>.
Then <math>F</math> has a maximal element with respect to set inclusion.}}

(Note that, strictly speaking, (1) is redundant since (2) implies the empty set is in <math>F</math>.) Note the above is a weak form of Zorn's lemma since Zorn's lemma says in particular that any set of subsets satisfying the above (1) and (2) has a maximal element ((3) is not needed). The point is that, conversely, Zorn's lemma follows from this weak form.<ref>{{harvnb|Halmos|1960|loc=§ 16.}} NB: in the reference, this deduction is by noting there is an order-preserving embedding 
:<math>s : P \hookrightarrow \mathfrak{P}(P)</math>
and that the "passage" allows to deduce the existence of a maximal element of <math>s(P)</math> or equivalently, that of <math>P</math> from the weak form of Zorn's lemma. The meaning of passage there was unclear and so here we gave an alternative reasoning.</ref> Indeed, let <math>F</math> be the set of all chains in <math>P</math>. Then it satisfies all of the above properties (it is nonempty since the empty subset is a chain.) Thus, by the above weak form, we find a maximal element <math>C</math> in <math>F</math>; i.e., a maximal chain in <math>P</math>. By the hypothesis of Zorn's lemma, <math>C</math> has an upper bound <math>x</math> in <math>P</math>. Then this <math>x</math> is a maximal element since if <math>y \ge x</math>, then <math>\widetilde{C} = C \cup \{ y \}</math> is larger than or equal to <math>C</math> and so <math>\widetilde{C} = C</math>. Thus, <math>y = x</math>.

The proof of the weak form is given in [[Hausdorff maximal principle#Proof]]. Indeed, the existence of a maximal chain is exactly the assertion of the Hausdorff maximal principle.

The same proof also shows the following equivalent variant of Zorn's lemma:<ref>{{harvnb|Halmos|1960|loc=§ 16. Exercise.}}</ref>

{{math_theorem|name=Lemma|math_statement=Let <math>P</math> be a partially ordered set in which each chain has a least upper bound in <math>P</math>.  Then <math>P</math> has a maximal element.}}

Indeed, trivially, Zorn's lemma implies the above lemma. Conversely, the above lemma implies the aforementioned weak form of Zorn's lemma, since a union gives a least upper bound.