I wish I could give you an exact figure. I normally shoot mine with a digital back, and I know I can shoot a couple thousand exposures, or close to, before having to change batteries. 4000 seems a bit much, but I'd think half that jives with my own experience. (I ran a portrait studio at a mall for 3.5 years from 2019-22, primarily using the 553ELX.) It actually does last a surprisingly long time on five AAs.
One thing you have to watch out for on the 553ELX is that on the battery chamber cover, there are metal tabs where the battery terminals make contact. Behind some of those there is some insulating material which keeps them from touching the metal of the cover itself, and creating a short. On both 553ELXes I have owned, one or more of those metal tabs pushed through the insulating material and shorted against the cover. You'll know when it's happened--your batteries will drain very quickly, and they'll get hot.
It's a very easy fix, even if not visually elegant: I just used a fine, flathead screwdriver to lift up the tips of the tabs where the short happens, and then cut really small squares of plastic-coated card stock (from the kinds of flyers people pass out for clubs) and slipped them under the metal tabs. Problem solved! You *can* get a new battery chamber cover from Hasselblad (well, at least you could several months back--they tried to sell me one when I sent the camera in for a service) but they want something like $275 for it. My solution (I actually got the idea from other people) works well--and is free.