Cartesian Product is Anticommutative

Theorem
Let $S, T \ne \O$.

Then:
 * $S \times T = T \times S \implies S = T$

Proof
Suppose $S \times T = T \times S$.

Then:

Thus it can be seen from the definition of set equality that $S \times T = T \times S \implies S = T$.

Note that if $S = \O$ or $T = \O$ then, from Cartesian Product is Empty iff Factor is Empty, $S \times T = T \times S = \O$ whatever $S$ and $T$ are, and the result does not hold.

Also see

 * Bijection between S x T and T x S