Ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ not isomorphic I'm doing this exercise:
Prove that ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ are not isomorphic ($\le_{lex}$ means lexigraphic order).
I don't know how to start (I know that to prove that ordered sets are isomorphic I would make a monotonic bijection, but how to prove they aren't isomorphic?).

Suppose there is an isomorphism $\phi:\mathbb{Q} \times \mathbb{N} \to \mathbb{N} \times \mathbb{Q}  $.
Let $(n,q) = \phi((0,0))$ and let $(a,b) = \phi^{-1}((n,q-1))$. Note that
we must have 
$a<0$ since $(n,q-1) < (n,q)$.
Note that we have $(a,b) < (a,b+1) < (0,0)$.
Note that there are no elements in $\mathbb{N} \times \mathbb{Q}$ between
$(n,q-1)$ and $(n,q)$
and since we must have
$(n,q-1) = \phi(a,b) < \phi(a,b+1) < \phi(0,0) = (n,q)$
we have a contradiction.
