How to prove that $x\cdot y\neq 0$ when $x\neq 0$ and $y\neq0$ via field axioms?

Question

How to prove that $x\cdot y\neq 0$ when $x\neq 0$ and $y\neq0$ via field axioms?

According to the field axioms, especially the Commutativity of multiplication it is $a\cdot b=b\cdot a$. Is that enough to disprove $x\cdot y=0$ hence proving that $x\cdot y\neq 0$

Answer 1

Suppose that $x\cdot y=0$ and $x\neq 0$, then $\frac1x$ exists and $\frac1x\cdot x\cdot y=\frac1x\cdot 0$, what is equivalent to $y=0$.

Answer 2

If $x\neq0$, then it as an inverse. So$$y=1.y=(x^{-1}.x).y=x^{-1}.(x.y)=x^{-1}.0=0.$$Now, it remains to be proved from the field axioms that you always have $x.0=0$.