formal languages – Show {0𝑚1𝑛|𝑚≠𝑛} is not regular

Try to express in natural language what $overline{L}$ contains; that is, what words $L$ doesn’t contain. Most obviously, it’s “words of the form $0^m0^n$, with $m = n$.” However, it also contains “words that are not of the form $0^m1^n$“, such as “$101010$“. That’s why the intersection with $0^*1^*$ is employed, to not bother with these words.

The demonstration is then possible because intersection and complementation are closed properties on the regular language. So if $overline{L} cap L(0^*1^*)$ is not regular, we know that $overline{L}$ must not be regular, so its complement, $L$, must not be regular.