While studying the compactness theorem, I was asked to prove equivalence of two statements of the theorem. I researched a bit and came across a question on this site that would hopefully answer my question, but the part that was unclear to the original author is unclear to me as well, and the accepted answer gave a rather short explanation, which I struggle to understand. The question: link
I will rephrase the question myself, starting off with the two statements.
Let $\Gamma$ be a set of propositional formulas, and let $\phi$ be a formula. If $\Gamma\models \phi$, then there is a finite subset $\Gamma_0\subset\Gamma$ such that $\Gamma_0\models\phi$.
If for each finite $\Gamma_0\subset\Gamma$ we have a model, then for $\Gamma$ we have a model.
After that, the author says
Now for $2\implies 1$, my book writes the following: Suppose 2. is true. Let $\Gamma$ be a set of propositional formulas, and let $\phi$ be a formula such that $\Gamma\models\phi$. Then there is no model for $\Gamma\cup\{\neg\phi\}$, and because of our assumption, there exists a finite $\Gamma_0\subset\Gamma$ such that there is no model for $\Gamma_0\cup\{\neg\phi\}$
The bolded part confuses me. If $\Gamma\cup\{\neg\phi\}$ doesn't have a model, then I can only conclude that it itself must have a subset which doesn't have a model either. I'd really appreciate some help understanding how the bolded claim came to be!