Unsatisfiable Set Union Formula 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 $\phi$ be a logical formula.


Then $\mathcal F \cup \left\{{\phi}\right\}$ is also $\mathscr M$-unsatisfiable.


Proof

This is an immediate consequence of Superset of Unsatisfiable Set is Unsatisfiable.

$\blacksquare$


Also see


Sources