Satisfiable Set Union Tautology is Satisfiable

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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.


Sources