Suppose that for every collection $A$ of non empty pairwise disjoint sets has a choice function. I need to prove that this implies the axiom of choice.
Let $S$ be a collection of sets. For every $B$ in $S$, define $S_B = {\{}{(x,B)|x\in B}{\}}$. The the set $S'={\{}S_D | D\in S{\}}$ is a set of pairwise disjoint sets, and therefore has a choice function $C$.
At this point, I wondered if the following is valid:
For every $B$, let $x_B$ be the unique element in the domain of ${\{}C(S_B){\}}$. Then define $K:S\to \bigcup S$, $K(B) = x_B\in B$ by definition, and therefore $K$ is a choice function on $B$.