Cartesian Product is Unique

Theorem

Let $A$ and $B$ be classes.

If there exists a cartesian product of $A$ and $B$, then it is unique.

Proof

Let $C_1$ and $C_2$ be cartesian products of $A$ and $B$.

Then by the cartesian product definition, for an arbitrary $a$:

$a \in C_1 \iff \exists x \in A: \exists y \in B: a = \tuple {x, y}$

$a \in C_2 \iff \exists x \in A: \exists y \in B: a = \tuple {x, y}$

By Biconditional is Transitive:

$a \in C_1 \iff a \in C_2$

By Axiom of Extension:

$C_1 = C_2$

Thus the cartesian product of $A$ and $B$ is unique.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 7$ Cartesian products