Proposition 1

If A$'$, B$'$ are non-empty sets, the relation A$' \times$ B$' \subset$ A $\times$ B is equivalent to "A$' \subset$ A and B$' \subset$ B".

In the first place, the relation $z \in A' \times B'$ is equivalent to "$z$ is an ordered pair and $\mathrm{pr}_1 z \in A'$ and $\mathrm{pr}_2 z \in B'$"; therefore, without any restriction on A$'$ and B$'$, the relation "A$' \subset$ A and B$' \subset$ B" implies

Conversely, let us first show that if $B' \neq \emptyset$ (without restriction on $A'$), the relation $A' \times B' \subset A \times B$ implies $A' \subset A$. Let $x$ be an element of $A'$; since $B' \neq \emptyset$, there is an object $y$ which is an element of $B'$; we have $(x, y) \in A' \times B'$, hence $(x, y) \in A \times B$, and consequently $x \in A$; thus $A' \subset A$. Similarly, if $A' \neq \emptyset$, the relation $A' \times B' \subset A \times B$ implies $B' \subset B$. Hence the result.