“God does not exist” turns out to make as much sense as “2 + 2 = 5.” You can string the symbols together, but you can’t get a concept out of that string.
As necessary, God is the logical forecondition of all contingent being. To see how this works, consider that for any x to happen at any time y, xy must have been possible before it came to pass. But because this is true for any x and any y, the possibility of every possible xy must be necessary. If there were no such possibility, there could never be any xy. None whatsoever: xy would in that case be simply impossible.
I.e.: if xy is possible, it is necessary that xy is possible.
Among other things, God is the possibility that xy. In his very essence, he furnishes all possibility. He is the forecondition of any and every being. It is in this sense that he is called the Ground of Being, or urgrund.
To say then that there is no Ground of Being is tantamount to saying that no being is possible, and that there is therefore no being of any sort. It is to say, “Nothing exists.” And the existence of that statement falsifies it.
It is hardly surprising then that no one can live as if it were true that God does not exist. We can say that he does not, and we can believe that we believe what we say, but we cannot live as if there were no such thing as being.