How to prove non-equality using Peano axioms?
Solution 1:
Inequalities are negations of equalities. That is, a claim like $1 \neq 2$ is really $\neg 1=2$. And negations are typically proven using a proof by contradiction. So: assume $1=2$ and derive a contradiction. You will find that you only need the first two Peano axioms to do this.
And if you have Modus Tollens, that works too: by axiom 2, you have $1=2 \to 0=1$, and by axiom 1, we have $\neg 0=1$, so by Modus Tollens, $\neg 1=2$