I'm a new student for Mathematical Logic and I am trying to solve this question:
In Hilbert proof system, prove or disprove the following: if $\Sigma\vdash A \lor B$ then $\Sigma \vdash A$ or $\Sigma \vdash B$.
I know that this is the same question in $\Sigma \ \vdash A \lor B \ \ $ but the answer contains things I haven't learnt. So please don't remark as duplicate.
I'm only at the beginning of system proofs and only know that if $\vdash A$ $\iff$ $A$ is tautology, I don't know if it's even relevant.
I am stuck as $\Sigma$ may be any set of propositions, but I think I need to disprove.