Modern large-scale industrial systems are complex in structure, and their health states are usually reflected by multiple indicators. The performance degradation of multiple indicators surpasses respective threshold, causing multiple system faults, and multiple failure modes such as redundancy, fusion, and competition among different fault combinations exist, introducing new challenges when predicting remaining useful life (RUL) of the system. This study aims to establish a unified multi-failure mode division framework and the RUL prediction model under different failure modes for multi-indicator systems. Firstly, combination relationship between different fault types caused by multi-indicators outweighing their threshold is analyzed, and different redundancy, fusion, and competition failure modes are defined. Next, the formal fault type definition and its remaining time before occurrence under different failure modes are provided. The distribution calculation model of remaining time for different fault types is derived. The corresponding system's RUL prediction model under multi-fault competition is established, and degradation modeling, parameter estimation, and RUL distribution calculation are performed using the state-space model. Eventually, the validity of the RUL prediction model according to multiple failure modes is verified by numerical experiments. Taking XJTU-SY bearing and C-MAPSS datasets as two examples, the applicability and feasibility of the given method are proved.