Subset of Satisfiable Set is Satisfiable
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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.
$\blacksquare$