Thanks to Elisha Otis, when the cables snap in modern elevators, the brakes act to keep the elevator from falling. The idea is similar to that of the air-brakes on trains, which are held open by pressure and brake the train when the pressure is released.
For somewhat the same reason as the electronic valve should not be called a “tube”, the elevator ought not to be called a “lift”. We have the former thanks to Britons; we don't fall to our deaths in the latter thanks to Americans.