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Automatic abstraction is an important component of modern formal verification flows. A number of effective SAT-based automatic abstraction methods use unsatisfiable cores to guide the construction of abstractions. In this paper we analyze the impact of unsatisfiable core minimization, using state-of-the-art algorithms for the computation of minimally unsatisfiable subformulas (MUSes), on the effectiveness of a hybrid (counterexample-based and proof-based) abstraction engine. We demonstrate empirically that core minimization can lead to a significant reduction in the total verification time, particularly on difficult testcases. However, the resulting abstractions are not necessarily smaller. We notice that by varying the minimization effort the abstraction size can be controlled in a non-trivial manner. Based on this observation, we achieve a further reduction in the total verification time.