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?