$Γ \models α $if and only if $Γ ∪ \{(¬α)\}$ is not satisfiable.

Question

Let $\Gamma$ be some set of well formed formulas and let $\alpha \in \mathrm{WFF}$. Prove the following statement:

• $\Gamma \models \alpha$ if and only if $\Gamma \cup \{(\lnot\alpha)\}$ is not satisfiable.

So far I have done this:

edit: whatever I did was wrong, so can you tell me what could be done?