"God doesn't Exist"- Claim or Conclusion

[Chas]

1: That God(s) exist is true.
2: Proposition 1 is not true.

Someone who considers proposition 1 true is a theist.
What do you call someone who considers proposition 2 true?

A (strong) atheist. A weak atheist doesn't yet consider either true.

I think not.  Proposition 2 is not strong atheism, it allows for agnostic atheism. 
Proposition 1 is not an existence claim, it is a claim that the existence claim is true.
Saying that it is false is saying that the existence claim has not been proved.