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?

This is one of those deceptively simple questions that is actually rather tough. Declaring $1 \neq 0$ seems rather cheap, but I can't think of a better way. In the hopes that it helps someone come to be a better answer, here's my attempt:
If $x = y$, then $x - y = 0$ and $x \div y = 1$. Obviously $1 - 0 = 1$ but that doesn't help us since maybe $1 = 0$. Nor does $0 \div 1 = 0$ help either. However, $1 \div 0$ is normally considered undefined, and attempts to define usually go for infinity.
Of course the big flaw in this whole argument is that $0 \div 0 \neq 1$ unless we explicitly defined it that way for the purpose of proving $0 = 1$.
