Logical deduction without completeness theorem

Question

I am just learning about deduction and I have to solve the following exercises

Let $\Sigma$ be a consistent set of propositions. Which of the following statements are necessarily true? Justify your answer

1. If $\Sigma\vdash (\phi \lor \psi)$ then $\Sigma\vdash \phi$ or $\Sigma \vdash \psi$.
2. If $\Sigma\vdash (\phi\iff \psi)$ then $\Sigma\vdash \phi$ or $\Sigma\vdash \psi$.
3. If $\Sigma\vdash \phi$ then $\Sigma \not\vdash \neg \phi.$

My attempt:

4. False. Counterexample, $\Sigma=\{\phi\lor \psi\}$. Clearly $\Sigma\vdash (\phi\lor \psi)$ but neither $\Sigma\vdash\phi$ nor $\Sigma\vdash \psi$.

5. Solved by @Daniel Schepler at comments

6. Follows directly from that $\Sigma$ is consistent.