Subset of Satisfiable Set 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 $\mathcal F'$ be a subset of $\mathcal F$.


Then $\mathcal F'$ is also $\mathscr M$-satisfiable.


Proof

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

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

Thus for every $\psi \in \mathcal F$:

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


Now, for every $\psi$ in $\mathcal F'$:

$\psi \in \mathcal F$

by definition of subset.

Hence:

$\forall \psi \in \mathcal F': \mathcal M \models_{\mathscr M} \psi$

that is, $\mathcal M$ is a model of $\mathcal F'$.


Hence $\mathcal F'$ is $\mathscr M$-satisfiable.

$\blacksquare$