Cartesian Product is Empty iff Factor is Empty

Theorem
$$S \times T = \varnothing \iff S = \varnothing \lor T = \varnothing$$.

Proof

 * Suppose $$S \times T \ne \varnothing$$:

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