Prove, axiomatically that $1$ does not equal $0$. I attempted proving this but came across many circular arguments. How would one prove such a claim without circular arguments?

Maybe we have an axiom that says $0$ is the additive identity and another axiom that says $1$ is the multiplicative identity, that is to say, $x + 0 = x$ and $x \times 1 = x$. Then, in order for $0 = 1$, we'd need to have $1 + 1 = 1$ and $0 \times 0 = 0$. But this implies that $1 \times 2 = 1$, contradicting the multiplicative identity axiom which requires $2 \times 1 = 2$. (Maybe we also need an axiom saying addition and multiplication are commutative).
