TITLE: How do I choose an element from a non-empty set?
QUESTION [16 upvotes]: Suppose I have a non-empty set $A$.

How do I choose an element $x\in A$?

More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is exactly one $x\in y$ such that $P(x,y)$ is true.
I have thought about this for a while without success and I am beginning to doubt such a formula is even possible. But in mathematical proofs it is quite common to "choose a fixed element" from a non-empty set. So how exactly do we achieve such a thing? Am I missing something?
Thank you in advance.

REPLY [2 votes]: David Hilbert introduced his $\tau$-operator (sometimes called $\epsilon$-operator) to extract such a choice function.  Foundationally this is somewhere between choice and global choice.  Bourbaki when they developed their own foundations choice to use Hilbert's $\epsilon$, getting them into no end of trouble as described in detail in the writings of Adrian Mathias; see e.g., this article.