[nnamtr]

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.

[twic]

Okay so if x is {potato, tomato}, what is the tuple for x, and how did you decide that?

[ambulancechaser]

A tuple (ordered pair commonly called) is `(x,y)`. The standard set theory definition underpinning this is that `(x,y)` is shorthand for the set of two elements `{ {x}, {x, y} }`. Where the "first coordinate" is the set with one element, and the "second coordinate" is the element in the set with two items that is not in the set with one item.

It's a standard definition that means we can talk about ordered pairs but still ground it in set theory.

[twic]

Yes, that's what nnamtr said in their comment, and it doesn't answer my question.