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

So I have the question: show “Show {0^𝑚1^𝑛|𝑚≠𝑛} is not regular”. I’ve already seen various proofs for this question, but they all have one step I don’t get.

They all take: “overline{L}∩(0∗1∗)” (overline{L} is the compliment of L) and show that it’s not regular. I don’t get why we can’t just take overline{L}. Because isn’t overline{L} = {0^𝑚1^𝑛|m=n} which is the same as {0^n1^n|n ≥ 0} which we know is not regular? What am I missing?