It's written in many resources that if we consider only finite sets, that choice axiom can be skipped and be proven from other ZF axioms.
I remember I've read the following explanation. If $A$ is a finite set, there is exists some natural $n, |A|=n$, therefore exists some bijection $f:\{1,\cdots,n\}\to A$. And we can choose an element from $A$ by choosing $f(1)$
But there is a problem with this explanation.
Actually there are $n!$ of such bijections. So how can we choose one $f$ of them without this axiom of choice?!
UPD: actually I can simplify my question heavily.
Let's consider one-point set $A, |A|=1$. How can I get an element from this set without using axiom of choice? We know that $A=\{x\}$ for some $x$. But we don't know how to obtain it. What allows us to just say Let's pick $x$ from $A$?