Subset of Satisfiable Set is Satisfiable

Theorem

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.

Let $\FF'$ be a subset of $\FF$.

Then $\FF'$ is also $\mathscr M$-satisfiable.

Proof

Since $\FF$ is $\mathscr M$-satisfiable, there exists some model $\MM$ of $\FF$:

$\MM \models_{\mathscr M} \FF$

Thus for every $\psi \in \FF$:

$\MM \models_{\mathscr M} \psi$

Now, for every $\psi$ in $\FF'$:

$\psi \in \FF$

by definition of subset.

Hence:

$\forall \psi \in \FF': \MM \models_{\mathscr M} \psi$

that is, $\MM$ is a model of $\FF'$.

Hence $\FF'$ is $\mathscr M$-satisfiable.

$\blacksquare$