Proof assistants like Isabelle/HOL provide the perfect opportunity to develop more than just one-off formalizations, but frameworks for developing new results within a given area. Completeness results for logical calculi often include bespoke versions of Lindenbaum’s lemma. In this paper, we mechanize an abstract, transfinite version of the lemma and use it to build witnessed, maximal consistent sets (MCSs) for any notion of consistency that satisfies a few requirements. We prove abstract results about when MCSs reflect the proof rules of the underlying calculi. Finally, we formalize a process for mechanically calculating saturated set conditions from a given logic’s semantics. This separates the truth lemma that connects MCS-membership and satisfiability into semantic and syntactic components, giving concrete proof obligations for each operator of the logic. To illustrate the framework’s applicability, we instantiate it with propositional, first-order, modal and hybrid logic examples. We mechanize strong completeness for each logic, even for uncountably large languages, proving that, if a formula is valid under a set of assumptions, then we can derive it from a finite subset.