Theorem 1

The relation

is true. In other words, for all X and all Y the relation "$z$ is an ordered pair and $\mathrm{pr}_1 z \in X$ and $\mathrm{pr}_2 z \in Y$" is collectivizing in $z$.

Let $A_y$ denote the set of all objects of the form $(x, y)$ for $x \in X$ (cf. §1, no. 6, criterion C53). Let R be the relation $z \in A_y$, which is equivalent to $(\exists x)(z = (x, y)$ and $x \in X)$. It is clear that the relation

is true, by virtue of S5 (Chapter I, §4, no. 2). It then follows from S8 that the relation $(\exists y)(y \in Y$ and R$)$ is collectivizing in $z$. But this relation is equivalent to $(\exists x)(\exists y)(y \in Y$ and $x \in X$ and $z = (x, y))$; hence the result.