Considering that $\Gamma$ is inconsistent if $ \Gamma \vdash ¬(\alpha \rightarrow \alpha) $ for some formula $\alpha$.
How to prove that if $ \Gamma $ is inconsistent, then $ \Gamma \vdash \beta $?
Do I need to assume that a proof of $ ¬(\alpha \rightarrow \alpha) $ exists and try to find $\beta$ using axioms? Is it possible?
List of axioms: