Completeness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
If a logical formula is a tautology, then it has a tableau proof.

That is:
 * If $\models \mathbf A$ then $\vdash \mathbf A$.

Proof
This is a corollary of the Extended Completeness Theorem of Propositional Logic:

Let $\mathbf H$ be a countable set of logical formulas.

Let $\mathbf A$ be a logical formula.

If $\mathbf H \models \mathbf A$, then $\mathbf H \vdash \mathbf A$.

In this case, we have $\mathbf H = \varnothing$.

Hence the result.

Also see
The Soundness Theorem of Propositional Logic in which it is proved that:
 * If $\vdash \mathbf A$ then $\models \mathbf A$.