Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a maximal element."
In proving the $\Rightarrow$ direction, he uses the following argument: Consider a nonempty subset $\Gamma'$ of $\Gamma$. Suppose that $\Gamma'$ has no maximal element. Then "by the axiom of choice" for any $\gamma \in \Gamma'$ we can select an element $\phi(\gamma')$ such that $\phi(\gamma')> \gamma'$. In that way we obtain a nonterminating ascending chain $\gamma' < \phi(\gamma')<\phi^2 (\gamma') <\cdots$, contradiction."
My question is: why is the axiom of choice necessary here? Don't we already have by hypothesis (no maximal element exists) a rule $\phi : \Gamma' \rightarrow \Gamma'$ to assign to each $\gamma'$ the element $\phi(\gamma')$?