Let $L_1$ be regular, $L_1 \cap L_2$ regular, $L_2$ not regular. Show that $L_1 \cup L_2$ is not regular or give a counterexample.
I tried this: Look at $L_1 \setminus (L_2 \cap L_1)$. This one is regular. I can construct a finite automaton for this: $L_1$ is regular, $L_2 \cap L_1$ is regular, so remove all the paths (finite amount) for $L_1 \cap L_2$ from the finite amount of paths for $L_1$. So there are a finite amount of paths left for this whole thing. This thing is disjoint from $L_2$, but how can I prove that the union of $L_1 \setminus (L_1 \cap L_2)$ (regular) and $L_2$ (not regular) is not regular?