Assume I have the following
(DIS) For every indexed family $\{A_i : i \in I \}$ there exists a family $\{B_i : i \in I \}$ of pairwise disjoint sets such that $B_i \subset A_i$ for all $i \in I$ and $\bigcup_{i \in I} B_i = \bigcup_{i \in I} A_i$.
and
(AC) For every family $x$ of non-empty, pairwise disjoint sets, there exists a set $z$ such that $|z \cap y | = 1$ for each $y \in x$.
I would like to show that $ZF \vdash (DIS) \rightarrow (AC)$ (that's another exercise in a book I'm currently reading).
So let's assume (DIS) and let $B_i$ be a family of non-empty pairwise disjoint sets. I don't see how to proceed. I am supposed to use (DIS) to construct $z$ such that $|z \cap B_i| = 1$ but my sets $B_i$ are already pairwise disjoint so what can I do? Thanks for your help.