I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah.
I have to do this and I don't know how to start.
Let $\Sigma_1 $ and $\Sigma_2$ be sets of sentences such that there is no model $M$ such that both $M\vDash \Sigma_1$ and $M\vDash \Sigma_2$.
Prove that there exists a sentence $\varphi$ such that:
Every model of $\Sigma_1$ satisfies $\varphi$ and every model of $\Sigma_2$ satisfies $\neg \varphi$.
I will be very grateful if you could give me a hint :) Thank you.