How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous proof. It would be nice if someone could direct me to a book containing such a proof.
EDIT: I am looking for literature where this is proved STRICTLY from the ZF axioms.
The proof is simple enough that you don't need a book. You just prove it by induction. The base case (one set) is trivial, and the inductive case is not much harder, using the fact that $$\prod_{i = 1}^{n+1} X_i \cong \left (\prod_{i=1}^n X_i\right ) \times X_{n+1}.$$