The following is a simple, straightforward Hartshorne-style ontological disproof of the existence of God. It uses modal logic: the modal operators <>x (it is possible that x) and x (it is necessary that x), and the modal rule <>~x -> ~x.
1. g -> g (basic in definition of God as a necessary entity).
2. <>~g (argued below).
3. ~g (from 2 by the rule stated above).
QED. ~g (modus tollens from 1 and 3).
Now, 2 is the point at which theistic attacks on this proof must be made. The burden of proof is 100% on the theist to show that, absolutely, ~<>~g. The atheist, meanwhile, merely has to maintain that it is possible that God does not exist, while the theist must show it is not possible at all. Since the major theistic arguments - teleological and cosmological - do not prove for certain ~<>~g, this argument suffices not only to say that we are not justified in believing in God, but that God doesn't exist at all. Until some theistic proof shows that it is absolutely necessary that God exist, and not just a supposedly likely proposition, the atheist is justified in saying that God does not, in fact, exist.
There are, of course, alternate routes; one is to abandon the necessity of God. However, this makes God a contingent being, and that idea does nobody good. So it can be dismissed. The other alternative is fideism - abandoning logic - which is just changing the rules when you can't win the game.