Theorem 7.6: (The Integrity of Multiplication of Natural Numbers) Let a and b be natural numbers. ab = 0 if and only if one of a or b are 0.

Proof. If either a or b are 0, then that number is equal to the empty set. In that case a x b is empty by Theorem 5.8, and then, by Theorem 6.6, ab is the number of elements in the empty set which is 0.

For the converse suppose that neither a nor b is 0. Then neither one is empty. Let

Then

and a x b is not empty, so ab is not equal to 0.