Compactness Theorem: If every finite subset of $\Gamma$ has a model, then $\Gamma$ has a model too.
I want to ask whether I proved correctly the corollary which says that If $\Gamma\vDash\phi$, then for some finite $\Delta\subseteq\Gamma$, $\Delta\vDash\phi$. Someone told me I could do a contrapositive proof, but wasn't sure how to do it.
My proof/reasoning:
If $\phi$ is a logical consequence of $\Gamma$, then $\Gamma\cup\{\lnot\phi\}$ is inconsistent. By the Compactness Theorem, there is a finite subset of $\Gamma\cup\{\lnot\phi\}$ that is inconsistent. So there is a finite subset $\Delta$ of $\Gamma$ such that $\Delta\cup\{\lnot\phi\}$
Thus, $\Delta\vDash\phi$