It's Countable Choice, and of course the essence of Countability (why it's our favorite cardinality) is that almost all of our intuition about finite numbers can be correctly applied to countable infinities.
Uncountable Choice (a set for every Real number, a choice from every set) is where thing ago off the rails and ZF looks substantially different from ZFC.
Can't you get countable choice with zf? Countable means there is a bijection to the naturals, so you can always pick the element that corresponds with 0.
