Satisfiable Set Union Tautology is Satisfiable

Theorem
Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$.

Let $\phi$ be a tautology for $\mathscr M$.

Then $\mathcal F \cup \left\{{\phi}\right\}$ is also $\mathscr M$-satisfiable.

Proof
Since $\mathcal F$ is $\mathscr M$-satisfiable, there exists some model $\mathcal M$ of $\mathcal F$:


 * $\mathcal M \models_{\mathscr M} \mathcal F$

Since $\psi$ is a tautology, also:


 * $\mathcal M \models_{\mathscr M} \psi$

Therefore, we conclude that:


 * $\mathcal M \models_{\mathscr M} \mathcal F \cup \left\{{\phi}\right\}$

i.e., $\mathcal F \cup \left\{{\phi}\right\}$ is satisfiable.