Could we show also that
" $M\cup N$ is finite $\implies$ $M$ and $N$ are finite sets "
as follows?

It holds that $M\subset M\cup N\Rightarrow |M|\leq |M\cup N|$. Since $|M\cup N|$ is finite it follows that |M| is finite.
The same for the set $N$ : $N\subset M\cup N\Rightarrow |N|\leq |M\cup N|$. Since $|M\cup N|$ is finite it follows that |N| is finite.

@Secret @JoeShmo The statement with the intersection is false, isn't it?

I mean " $M\cap N$ is finite $\implies$ $M$ and $N$ are finite sets " is not true, or not? To find a counterexample do we take one of M or N as an infinite set and the other one as a finite example that contains just some elements of the infinite set?