Unsatisfiable Set minus Tautology 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 $\phi \in \FF$ be a tautology.

Then $\FF \setminus \set {\phi}$ is also $\mathscr M$-unsatisfiable.

Proof

Suppose $\FF \setminus \set {\phi}$ were satisfiable.

Then by Satisfiable Set Union Tautology is Satisfiable, so would $\FF$ be, because:

$\FF = \paren {\FF \setminus \set {\phi} } \cup \set {\phi}$

by Set Difference Union Intersection and Intersection with Subset is Subset.

Therefore, $\FF \setminus \set {\phi}$ must be unsatisfiable.

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Theorem $2.47$
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.15$