I'm taking a lecture in Mathematical Logic for the first time, and the following is a step in the proof of the completeness theorem that I can't work out.
Let $\Gamma^*$ be maximally consistent. For propositional formulas $\phi,\psi$, we have that $v(\phi)=1 \iff \phi \in \Gamma^*$ and $v(\psi)=1 \iff \psi \in \Gamma^*$. Prove that $v(\phi \vee \psi)=1 \iff \phi \vee \psi \in \Gamma^*$.
I think I have managed to do this using $\vee$ introduction and elimination rules. The problem is, those rules were not introduced in my lectures (it was noted that $\land, \rightarrow$, $\neg$ are adequate thus rules for other connectives would not be explained, nor allowed on exams), and also for this particular problem I'm not allowed to use them.
How can I prove the above without using ($\vee$I) and ($\vee$E)? This comes up as a step in proving the completeness theorem, so we are not allowed to use it. But to be honest, I'm unsure as to whether or not we can even use that "$\land, \rightarrow$, $\neg$ are adequate thus it is not necessary to introduce rules for other connectives" without being able to use the completeness theorem. Does anyone know how to do this? (Also, is it possible to use/conclude "$\land, \rightarrow$, $\neg$ are adequate thus it is not necessary to introduce rules for other connectives" without using the completeness theorem?)