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


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 $2.15$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.