Theorem   The well-ordering principle implies the axiom of choice.

Proof. Let $C$ be a collection of nonempty sets. Then $\displaystyle \bigcup_{S \in C} S$ is a set. By the well-ordering principle, $\displaystyle \bigcup_{S \in C} S$ is well-ordered under some relation $<$ Since each $S$ is a nonempty subset of $\displaystyle \bigcup_{S \in C} S$ each $S$ has a least member $m_S$ with respect to the relation $<$

Define $\displaystyle f \colon C \to \bigcup_{S \in C} S$ by $f(S)=m_S$ Then $f$ is a choice function. Hence, the axiom of choice holds.