Using the five Peano axioms of first-order theory arithmetic, one can prove '1+1=2', '2+3=5' or '342+637=979' (don't try the last one!). However, my question is how to prove a non-equality such as "1+1≠3" and "1≠2" using Peano axioms.
For the second case, can we say by definition of '1' and '2', it is "1≠2"? I have no idea how to proceed with cases like the first one. My instructor gave me a hint to use modus tollens.
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, $\neg 1=2$