Unsatisfiable Set minus Tautology is Unsatisfiable

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 $\phi \in \mathcal F$ be a tautology.

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

Proof
Suppose $\mathcal F \setminus \set {\phi}$ were satisfiable.

Then by Satisfiable Set Union Tautology is Satisfiable, so would $\mathcal F$ be, for:


 * $\mathcal F = \paren {\mathcal F \setminus \set {\phi} } \cup \set {\phi}$

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

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