I have a question about a gap in lemma. First how things were defined in the course I'm taking (I'm sorry to be making the readers going through this list of definitions, but I don't know how to make it shorter):
Let $\sigma $ be a signature and $T$ a $\sigma $-theory. We have defined this theory to be satisfiable if there is a model $M$ such that for every sentence of the theory $M$ satisfies the sentences. We have defined sentences to be provable if they belong to the smallest set satisfying a list of properties and universal if for every model $M$ they are satisfied. Further on, we have defined the theory to be contradiction free if there are no sentences $\alpha_1,\ldots,\alpha_n \in T$ such that $\neg ( \alpha_1 \wedge \ldots \wedge \alpha_n)$ is provable.
Then we proved that every provable sentence is universal and then it was mentioned (without a full proof) that by the previous proposition every satisfiable theory is contradictions free. My question is: How can the proposition that every provable sentence is universal be used to prove that ?