Using general relativity (which has plenty of equations), and a little borrowed knowledge from other fields (to describe star collapse), you can show that black holes should exist. But unfortunately there are no proofs in physics, just experimental and observational evidence. That being said, the observational evidence of the existence of the black holes has been extremely good. For example, by looking at the movement of the stars in the galactic core we’ve determined that they must be orbiting a tiny, invisible object with several million Suns worth of mass. So Sagittarius A* (the super-massive object in the center of our galaxy) is almost definitely a black hole, but it hasn’t been (and never will be completely, absolutely, and totally) proven that black holes exist.