In recent years, IC3 has enjoyed wide adoption by academia and industry as an unbounded model checking engine. The core algorithm works by learning lemmas that, given a safe property, eventually converge to an inductive proof. As such, its runtime performance is heavily dependent upon 'pushing' (or 'promoting') important lemmas, possibly by discovering additional supporting lemmas. More recently, Quip has emerged to be a complementary extension behind the reasoning capabilities of IC3 as it allows it to target particular lemmas for pushing. This also raises the following question: which lemmas should be promoted? To that end, this paper extends the reasoning capabilities of IC3 and Quip using special SAT queries to find support sets that represent fine-grained information on which lemmas are required to push other lemmas. Further, this paper presents an IC3-based algorithm called Truss (Testing Reachability Using Support Sets) that uses support sets to identify sets of lemmas that may be close to forming an inductive proof. The set is targeted for promotion as a cohesive unit. If any of the lemmas cannot be promoted, the entire set is abandoned and a new set excluding that lemma is found. In the presented framework, there are two reasons why a lemma cannot be promoted: either because it blocks a known reachable state (in which case, the lemma is permanently marked as bad), or because lemma promotion exceeds a specified amount of effort (in which case the lemma is temporarily marked as ugly). Intuitively, the proposed approach allows the algorithm to construct a proof more quickly by focusing on the important yet easily-pushed lemmas. Experiments on the HWMCC'15 benchmark set show a significant improvement against existing practices. Compared to Quip, our algorithm solves 17 more problem instances and it offers an impressive 1.77× speedup.