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

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

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


 * $\mathcal F = \left({\mathcal F \setminus \left\{{\phi}\right\}}\right) \cup \left\{{\phi}\right\}$

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

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