Need help with a proof that if $xy=0$ then $x=0$ or $y=0$

I need to prove the following statement: if $xy=0$ then $x=0$ $or$ $y=0$.

My first idea was to prove the negation of the original problem if $x\neq0$ and $y\neq0$ then $xy\neq0$ but I still don't know how to prove it rigorously.

Solution 1:

Hint: Suppose that $x\ne 0$ and $xy=0.$ Since $x\ne 0,$ then there is a very special real number closely tied to $x$ with regard to multiplication. Check your axioms, and see if you can figure out what it could be. What will this real number allow us to do, since $1y=y$ and $z\cdot 0=0$ for all $z$?

Solution 2:

Suppose $xy=0$ and suppose $x\ne0$ and $y\ne 0$. Then there exists $x^{-1}$ and $y^{-1}$. Now, you should be able to derive a contradiction.