injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto (m+n)^{\max\{m,n\}}$, where $\mathbb{N}$ denotes the natural numbers.

How to prove that this map is injective? It should be elementary. We might be overlooking something trivial.

Thanks!

Edit As it was pointed out, it is not an injection by easy reasons. Thanks a lot! I was just overcomplicating things. But what if we restrict the map to the set of pairs $(m,n)$ such that $m>n$?

It is not an injection since $m+n=n+m$ and $\max(m,n)=\max(n,m)$.

Is not. For example $(3,2)$ and $(2,3)$ are both mapped to $5^3$.

A simple injection is given by $(m,n) \mapsto 2^m 3^n$.