Cartesian Product is Empty iff Factor is Empty

Theorem

 * $$S \times T = \varnothing \iff S = \varnothing \or T = \varnothing$$.

Thus:
 * $$S \times \varnothing = \varnothing = \varnothing \times T$$

Proof
$$ $$ $$ $$ $$

So by the Rule of Transposition $$S = \varnothing \or T = \varnothing \iff S \times T = \varnothing$$.