Cartesian Product with Proper Class is Proper Class
Theorem
Let $A$ be a proper class.
Let $B$ be a class which is not empty.
Then the Cartesian product $\paren {A \times B}$ is a proper class.
Proof
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Aiming for a contradiction, suppose that $\paren {A \times B}$ is small.
By Domain of Small Relation is Small, the domain of $\paren {A \times B}$ is small.
Since $B \ne \O$, Nonempty Class has Members shows that $\exists y: y \in B$.
The domain of $\paren {A \times B}$ is the collection of all $x \in A$ such that $\exists y: y \in B$.
The domain of $\paren {A \times B}$ is $A$.
Therefore, $A$ is small, contradicting the fact that it is a proper class.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.9 \ (2)$