All formulas and statements are expressed in first-order logic. It has no such phrase as "pick". But picking (in the axiom of choice) means: Finding a function that maps every set to an element of this set.

What is a function: It's a set of tuples where every tuple (x, y) means: x is mapped to y. Of course there must be exactly one tuple for every "x".

(Tuples btw are usually a short notion for {{x}, {x,y}}. They exist always by the axiom of pairing. By the axiom of extensionalty, they fulfill the universal property of pairs, i.e. (x, y) = (z, w) iff x = z and y = w.)

Assume x is a single nonempty set. Picking means: Finding a function {x} -> x. But this is easy: ∃y(yϵx) because x is nonempty, and then we can build our function, which consists just of the single tuple (x, y).

It also works for finite sets of sets since you can do it manually n times, the formula just gets longer.