Unsatisfiable Set Union Formula is Unsatisfiable

Theorem

Let $\LL$ be a logical language.

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

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

Let $\phi$ be a logical formula.

Then $\FF \cup \set \phi$ is also $\mathscr M$-unsatisfiable.

Proof

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

$\blacksquare$

Also see

• Superset of Unsatisfiable Set is Unsatisfiable

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Theorem $2.46$
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.15$