There are two descriptions of axiom of choice that are considered equivalent.
The first one goes like this:
Let $({X_i})_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.
Then there exists an indexed family $({x_i})_{i \mathop \in I}$ such that $\forall i \in I: x_i \in X_i$
The second one goes like this:
That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.
I don't see how the second one is equivalent to the first one. Can someone illustrate with an example that both the descriptions mean the same thing?