Proof involving Peano axiom

Question

Prove that $n \ne n+1 $ for all $ n \in \mathbb N.$

How do I prove this statement using only Peano axioms?

Answer 1

You'll have to use induction. Here is a formal proof in Fitch, which has Induction as a built in rule. Note that $s(x)$ is the successor function, i.e. $s(x)=x+1$