Prove using axioms that $1 \neq 0$ - checking

Question

Let $1=0$ we know that $x \cdot y=0$ if $x=0$ or $y=0$ so

$1(1-1)=0 $ if $1=0$ or $1=1$ but $1=1$ is contradiction since we assume that $1=0$ so $1 \neq 0$

Is my prof correct ? If not how should it look like ?