Consequences of Axiom of Choice

Question

In the proof of Zorn's lemma by Jonathan Lewin published in AMM in 1991, the following reasoning appears:
Suppose every chain of a set $\mathcal{P}$ has a strict upper bound in $\mathcal{P}$(an element strictly greater than all the elements ).Then by Axiom of choice, there exists a function $f$ mapping every chain in $\mathcal{P}$ to its strict upper bound.

I am not sure how is this derived directly from Axiom of Choice, which only states there exists some function mapping every chain to its element. Nothing is said about choosing a particular element, and let alone the fact the upper bound is not even an element of the set. It might be a simple consequence of Axiom of Choice, but this is a new field for me so I would appreciate a detailed explanation.

Answer 1

Assume that every chain has a strict upper bound.

• Therefore given a chain $C$, the set $U_C=\{x\mid x\text{ is a strict upper bound of }C\}$ is non-empty.
• Therefore the family $\{U_C\mid C\text{ is a chain}\}$ is a set of non-empty sets.
• Therefore it admits a choice function, $f$, that is $f(U_C)\in U_C$.
• Therefore the function $F(C)=f(U_C)$ is a function mapping a chain to a strict upper bound.