Is that not the axiom of finite choice, which holds in ZF set theory without AC?
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.