Note that $\in_{cw}$ is the canonical well-ordering of $\mathbb{N}\times\mathbb{N}$.
That is, $(m_1,n_1)\in_{cw}(m_2,n_2)$ if and only if
either $\max\{m_1,n_1\}<\max\{m_2,n_2\}$,
or $\max\{m_1,n_1\}=\max\{m_2,n_2\}$ and $m_1<m_2$,
or $\max\{m_1,n_1\}=\max\{m_2,n_2\}$ and $m_1\le m_2$ and $n_1<n_2$.
I'm trying to prove it in this way:
Define $f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ by:
$f(m,n)=m^2+m+n$ if $n\leq m$
$f(m,n)=n^2+m$ if $m<n$
Clearly, f is a bijection. It remains to show that $(m_1,n_1)\in_{cw}(m_2,n_2)$ if and only if $f(m_1,n_1)<f(m_2,n_2)$. How to do this?