Cartesian Product is Unique

Theorem
Let $A$ and $B$ be classes.

There exists a unique cartesian product of $A$ and $B$.

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.