Why didn't Voldemort die....?

So in Harry Potter and the Deathly Hallows, once Neville kills Nagini, supposedly all of the Horcruxes are gone. Diary, Locket, Cup, Diadem, Harry, Nagini, Ring. Well if on the night in Godric's Hollow when Voldemort tried to kill Harry and was defeated, does that mean that part of his soul died that night, and his Horcruxes kept him alive? So my question is, if the part of soul that is still inside Voldemort would have died on the night in Godric's Hollow, shouldn't he have automatically died in The Battle of Hogwarts when Neville killed the snake?