If you are looking for a Scientific proof, I doubt whether it exists.
on the contrary if you understand them as energy (separated energy) then they do exists. We know mind exist, but we don't know what it is. Where is mind positioned in the body. We are not even sure if the Mind is in the body or out side enveloping the body.What happens to mind on death is still a mystery.
Yes our soul does exist. various religious books tell about life and after life and soul. Garudapurana is one among those books that tells us more about death and life after.
It is up to you to read and investigate.