2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229841The statement of the title is proved. It implies that under physically reasonable conditions, spacetimes which are free from singularities are necessarily stably causal and hence admit a time function. Read as a singularity theorem it states that if there is some form of causality violation on spacetime then either it is the worst possible, namely violation of chronology, or there is a singularity. The analogous result: "Non-totally vicious spacetimes without lightlike rays are globally hyperbolic" is also proved, and its physical consequences are explored.20 pages, 6 figuresGeneral Relativity and Quantum Cosmologytext