I have recently started to learn a propositional logic from a textbook. In one exercise I should make a proof of the following statement:
"If T is a set of premises and A is a premise, than T ⊨ A and T ⊨ ¬A, iff T is not satisfiable."
To me, it is obvious that this statement is true because I know that (T ⊨ A) ∧ (T ⊨ ¬A) is a contradiction and I also know that being a contradiction means that it is always false, which means that it is not satisfiable. Yet, I don't know if I can call this argument a proof of that statement.
How should I make a proof of that statement so that my teacher of mathematical logic would accept it as a proof?