How to show if $\Gamma\models\bot$, then $\Gamma$ is not satisfiable?

Question

I am struggling to show that for any set of sentences $\Gamma$, if $\Gamma\models\bot$, i.e. $\Gamma$ entails $\bot$, then $\Gamma$ is not satisfiable, i.e. for any structure $M$, $M\not\models\Gamma$.

My idea is to prove by contradiction:

Assume to the contrary that $\Gamma\models\bot$ and $\Gamma$ is satisfiable. Then there is some structure $M$ s.t. $M\models\Gamma$. Since $\Gamma\models\bot$, $M\models\bot$...

I am not sure what to do next.

Answer 1

First, suppose that $\Gamma \models \bot$, then for any $p, \Gamma \models p \land \neg p$ (why?).

Now let M be any model of $\Gamma$. Then apply the definition of satisfaction to obtain a contradiction.