TITLE: Cambridge Tripos 2015: Zorn's Lemma Proof Requiring Axiom of Choice
QUESTION [9 upvotes]: I was doing a Cambridge Tripos, and the task was to prove Zorn's Lemma and show where the Axiom of Choice was assumed and how it fit into the proof. Reading a paper, I found this:

To prove Zorn’s lemma, it will be convenient to assume that we have a a [sic] “successor operation” on $X$ [a partially ordered set], denoted $x\mapsto x^+$, such that $x^+ > x$ if $x$ is not maximal, and $x^+ = x$ if $x$ is maximal. (The axiom of choice guarantees that there is indeed such a function.)

My question is how the Axiom of Choice guarantees the existence of this function.

REPLY [7 votes]: I presume $X$ is a partially ordered set. For each nonmaximal
$x\in X$, then $A_x=\{y\in X:y> x\}$ is nonempty. But if $x$ is maximal, define $A_x=\{x\}$.
By AC then there is a choice function $f:X\to X$ with $f(x)\in A_x$. You could write $x^+$ for $f(x)$.