In this exploratory study, we investigate undergraduate students' engagement with generative Artificial Intelligence (genAI) in proving mathematical statements. We selected six mathematical statements to conduct interviews with three students. We present the emergent framework, Students' Interactive Proving Experience with AI (SIPE-AI), which explains the processes of students' use of genAI in their proving and the factors influencing these processes. Our findings identify three factors that shape students' use of genAI: conceptions of proof, conceptions of genAI, and ethical considerations. The results suggest a need to guide undergraduate students in critically engaging with genAI tools, rather than passively accepting their outputs. We also discuss the implications of these findings for enhancing undergraduate mathematics education by fostering informed and critical use of genAI in mathematical proving.