The following excerpt is from Ethan Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics (2nd ed, 2011 : page 121) and concerns one of the motivations for introducing the axiom of choice (AC). Initially, the author explains that for a finite collection $\Gamma$ of nonempty finite sets $\Gamma_i$, we can choose a single element $\gamma \in \Gamma_i$ sequentially for all $i \in I$ where $I$ is an indexing set, and eventually we will have a new set $\Lambda$ which contains those elements $\gamma$ in a finite number of steps. The problem of choosing a single element sequentially arises when the collection is infinitely large:
If we want to choose one element from each set in an infinite family of nonempty sets, we need to make the choices simultaneously. Such a simultaneous choice is not something we could physically do, and the ability to do so mathematically does not follow from the [ZF] axioms of set theory. Therefore, we need an additional axiom to guarantee our ability to make such choices, and that axiom is the Axiom of Choice
Since the discussion of AC was introduced prior to the concept of a function, the author defines AC in terms of sets:
Let $x$ be a set. Suppose that if $y,w \in x$, then $y \neq \varnothing$ and $y \cap w = \varnothing$. Then there is a set $z$ such that if $y \in x$, then $y \cap z$ contains a single element.
Okay, now that I have laid the foundation, here is my question: What in the above definition entails that the elements are simultaneously chosen when $x$ is an infinite collection of nonempty sets?