Product of Nonzero Integers Is Nonzero

We have the following statement: If $a,b\neq 0$, then $ab\neq0$.

You can try, but you can't. How do we know? Well, let's try to prove that it cannot be proved. We can do this by showing that if we did prove it, a contradiction will occur.

So we do something really clever. Note that all of these properties apply also to $\mathbb{Z}/6\mathbb{Z}$. Thus, if we prove it in $\mathbb{Z}$, then we would have proven it for $\mathbb{Z}/6\mathbb{Z}$ too. However, we know that in $\mathbb{Z}/6\mathbb{Z}$, we have $2\cdot3=0$. Thus, it is not provable in $\mathbb{Z}/6\mathbb{Z}$, and consequently it is not provable in $\mathbb{Z}$ as well.