Proof of: $m = n = 0 \iff m + n =0$ $n,m \in N$

Question

I need to prove only one way,

I know that $m + n =0$ with $n,m \in \mathbb{N}$ $\implies m = n = 0$

only using sum axioms $\forall n \in \mathbb{N}, n+0=n$

Answer 1

$\mathbb N=\{0,1,2,3,\ldots\} \implies m,n\geq0.$ Suppose $m\neq0$, then $m\geq1$ so $m+n\geq1$ and $0\geq1$, contradiction.