Pumping Lemma Puzzle: Sipser 1.51b

(Sipser 1.51b or 1.46b) Prove that $L = \{0^m1^n \mid m \neq n \}$ is not regular. Use the pumping lemma in addition to the closure of the regular languages under Union, Intersection, and Complement.

Hint: First, find a language $L’$ that you know is non-regular (or is easy to show is non-regular via the pumping lemma) that is closely related to $L$. Then use the complement, or the union/intersection with a known regular language, to show that if $L$ were regular, $L’$ would be regular too! This should set you up for a contradiction.