I have to show: $$xy\neq0 \Leftrightarrow x\neq0 \wedge y \neq0 $$
I think I can "simplify" it to this: $$xy=0 \Leftrightarrow x=0 \vee y=0 $$
Since $a\cdot0=0$ is an proven theorem, I can show: $$x=0 \vee y=0 \rightarrow xy=0 $$
But that is just one of the directions. How can I use the field axioms to show this theorem?