Prove that if $\Gamma$ is inconsistent, then $\Gamma$ doesn’t have a model.

Question

The task is to prove that If set of sentences $\Gamma$ is inconsistent, then $\Gamma$ doesn't have a model. It is a corollary of Soundness Theorem: If $\Gamma\vdash A$, then $\Gamma\models A$. Could anyone explain how to prove it?

Answer 1

Suppose $\Gamma$ is inconsistent, then for some formula $\phi$, $\Gamma\vdash\phi,\neg\phi$. By the soundness theorem, $\Gamma\vDash\phi,\neg\phi$. Any model for $\Gamma$ thus satisfies $\phi$, but then it cannot satisfy $\neg\phi$, which is a contradiction, so $\Gamma$ has no model.