Proposition 7

Let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ be a family of cardinals. Then $\mathbf{P}_{\iota \in \mathrm{I}}\, \mathfrak{a}_\iota \neq 0$ if and only if $\mathfrak{a}_\iota \neq 0$ for all $\iota \in \mathrm{I}$.

This is merely a translation of the condition that a product set should be non-empty (Chapter II, § 5, no. 4, Proposition 5, Corollary 2).