Assign a primitive proposition true exactly when it belongs to a maximal consistent set. An induction on formulas proves that truth agrees with membership: maximal consistency gives the negation dichotomy, and deductive closure gives the implication truth table. This builds a model of every consistent propositional theory.
Consistency is preserved under unions of chains because proofs use finitely many assumptions. A maximal consistent extension then gives a valuation through its truth lemma, proving propositional completeness.

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  1. Completeness theorem for propositional logic
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