I know perfectly well what an axiom is and I still don't think your "god exists" axiom is useful for anything than stating that "something exists". You have to add more axioms.

You may prove many things if you add different axioms:

1)
Ok. I add the axiom A = "for all a,b if a is god and b is god, then a = b". From that it is trivial to derive the theorem "there is only one that is god".

2)
Now instead of the axiom A, I add the axiom B = not A. From that it's trivial to derive the theorem "there may be two or many that are god"

3)
Let's check what happens if we add A and B. Well from that you can derive a contradiction and it's a logic truth that from a contradiction everything can be derived. One example is the theorem "there is no god".