Prove: For any $x,y \in \mathbb{N}, y \neq x+y$.
I'm only suppose to use the Peano axioms as defined here http://aleph0.clarku.edu/~djoyce/numbers/peano.pdf and the properties of addition in $\mathbb{N}$.
I've already proved most of the properties of addition excepting this one and I don't know how to approach to this.
Hints: Take an arbitrary $x\in \mathbb N$ and prove the statement $\forall y\in \mathbb N(x\neq x+y)$ by induction. You'll have to use the definition of $+$, the injectivity of the successor function and the fact that $1$ is not the successor of any number.