# 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 $\left({ A \times B }\right)$ is a proper class.

## Proof

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Suppose, to the contrary, that $\left({ A \times B }\right)$ is small.

By Domain of Small Relation is Small, the domain of $\left({ A \times B }\right)$ is small.

Since $B \ne \varnothing$, Nonempty Class has Members shows that $\exists y: y \in B$.

The domain of $\left({ A \times B }\right)$ is the collection of all $x \in A$ such that $\exists y: y \in B$.

The domain of $\left({ A \times B }\right)$ 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*: $\S 6.9 \ (2)$