Show $\{0^1^|≠\}$ 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 &quot;words of the form $0^m0^n$, with $m = n$.&quot; However, it also contains &quot;words that are not of the form $0^m1^n$&quot;, such as &quot;$101010$&quot;. 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.