Cartesian Product is Anticommutative

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

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 $$S = \varnothing$$ or $$T = \varnothing$$ then, from Cartesian Product Null, $$S \times T = T \times S = \varnothing$$ whatever $$S$$ and $$T$$ are, and the result does not hold.