We present a model checking approach for axiomatic memory consistency models written in the language CAT. It can prove properties of single memory models like the monotonicity of barriers, and also compare models like TSO and ARM8. To achieve this expressiveness, our approach supports the full rational fragment of CAT. We not only support the Kleene star operation, composition, and union to define relations in a memory model, but also, for the first time, intersection, converse, and the construction of relations from sets. Our model checking approach for memory consistency models is logical in nature: we formulate the problem as satisfiability in a logical theory of relations. Our technical contribution is then a new theory solver that is sound, complete, and optimal from a complexity point of view. At the heart of our solver is a cyclic proof system that can detect invariants on-the-fly. This allows us to terminate early not only in the case that a model has been found, but also in the case of unsatisfiability. Our solver is easy to implement and easy to combine with heuristics. We have implemented a combination with counterexample-guided abstraction refinement, and exercised the prototype on a number of benchmarks — with promising results.