The verification of strategic abilities of autonomous agents is a key subject of investigation in the applications of formal methods to the design and certification of multi-agents systems. In this contribution we propose a novel approach to this verification problem. Inspired by recent advances, we introduce a translation from Alternating-time Temporal Logic (ATL) to First-order Logic (FOL). We show that our translation is sound on a fragment of ATL, that we call ATL-live, as it is suitable to express liveness properties in MAS. Further, we show how the universal model checking problem for ATL-live can be reduced to semantic entailment in FOL. Finally, we prove that ATL-live is maximal in the sense that if any other ATL connective is added, non-FOL reasoning techniques would be required. These results are meant to be a first step towards the application of FOL reasoners to model check strategic abilities expressed in ATL.