Corollary 2

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets. Then $\prod\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota = \emptyset$ if and only if there exists $\iota \in \mathrm{I}$ such that $\mathrm{X}_\iota = \emptyset$.

If we have $\mathrm{X}_\iota \neq \emptyset$ for each $\iota \in \mathrm{I}$, then it follows from Corollary 1 that $\prod\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota \neq \emptyset$; conversely, if $\prod\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota \neq \emptyset$, the relation $\mathrm{pr}_\alpha\Big(\prod\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota\Big) \subset \mathrm{X}_\alpha$ shows that $\mathrm{X}_\alpha \neq \emptyset$ for each $\alpha \in \mathrm{I}$.

Hence, if we have a family $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ of non-empty sets, we may introduce (as an auxiliary constant) a function $f$ with domain I such that $f(\iota) \in \mathrm{X}_\iota$ for all $\iota \in \mathrm{I}$. In practice one says : "take an element $x_\iota$ in each $\mathrm{X}_\iota$". Intuitively, we have thus "chosen" an element $x_\iota$ in each set $\mathrm{X}_\iota$; the introduction of the logical sign $\tau$ and the criteria governing its use absolve us from the necessity of formulating an "axiom of choice" to legalize this operation (cf. Summary of Results, §4, no. 10).