I want to proof that the set of spectra is closed under union and intersection. I know that it works somehow by assuming that the languages of the two formulas $\phi_1$ and $\phi_2$ are disjoint and considering $\phi_1 \vee \phi_2$ and $\phi_1 \wedge \phi_2$, but I have no idea why we can assume that they are disjoint without loss of generality.
I'd appreciate any help for this exercise.