  It is to be noted that AC1 and CAC for finite collections of sets are both provable (by induction) in the usual set theories. But in the case of an infinite collection, even when each of its members is finite, the question of the existence of a choice function or a transversal is  problematic[4].  For example, as already mentioned, it is easy to come up with a choice function for the collection of pairs of real numbers (simply choose the smaller element of each pair). But it is by no means obvious how to produce a choice function for the collection of pairs of arbitrary sets of real numbers.