  The 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.
