In many mathematical proofs dealing with mathematical objects, we begin by saying something along the lines of: Let $X$ be a such and such mathematical object. Let $y \in X$.
And then we proceed to exploit the properties of $X$ to prove what we want using $y$. This is so commonplace, and what I would like to know is:
How is this different from assuming the axiom of choice? We are inherently assuming that we can choose an element from some sort of set based object. Often, this element $y$ is taken to be arbitrary, but this is often said near the end of the proof. Is it important that we are doing this with one "set based object", instead of a set of "set based objects"?
Thank you