I am reading Lee's Introduction to Smooth Manifolds. In Theorem 2.17, it is claimed that if $M$ and $N$ are smooth manifolds with boundary, and $F:M \rightarrow N$ is a diffeomorphism, then $$F(\partial M)=\partial N,$$ where $\partial M$ and $\partial N$ are the boundaries of $M$ and $N$, respectively. Maybe I'm missing something, but is there a non-trivial conclusion that we can draw from this? That is:
Claim: If $M$ and $N$ are diffeomorphic, then $\partial M= \emptyset $ if and only if $\partial N = \emptyset$.
Does this follow, or am I missing something?