Superset of Unsatisfiable Set is Unsatisfiable

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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 $\mathscr M$-unsatisfiable set of formulas from $\mathcal L$.

Let $\mathcal F'$ be a superset of $\mathcal F$.


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


Proof

By assumption, $\mathcal F$ is unsatisfiable.


Suppose now $\mathcal F'$ were satisfiable.

Then it would follow from Subset of Satisfiable Set is Satisfiable that $\mathcal F$ were also satisfiable.


We conclude that $\mathcal F'$ must be unsatisfiable.

$\blacksquare$