Prove or disprove: If $A$ is regular and $A ∩ B$ is not regular, then $B$ is not regular.

Question

I suspect the Myhill–Nerode theorem may come into play, but not certain. If this was a union instead of an intersect, I'd be almost 100% sure it was true. I'm relatively confident that this statement is true and thus, is to be proven, but not certain how to formally prove it. Of course, I could be wrong and it's false. Anything to help me better understand this would be greatly appreciated.

Answer 1

It simply follows from the known fact that the intersection of two regular languages is regular. Thus if $B$ were regular, then $A \cap B$ would be regular, a contradiction.