If two languages L1 and L2 are regular than so is their intersection.
If follows that, for two languages L1 and L2, if L1 intersection L2 is not regular and L1 IS regular than L2 is not regular.
I am having a hard time proving this. Since I am dealing with regular languages, I decided to build DFA's for them to help my proof. However, this does not help with my proof if I am supposed to prove the irregularity of a language. How do you prove a language is not regular if you cannot construct a DFA/NFA for them? Is pumping lemma the only way?