In fact what I say is a little less strong: you can’t have absolute nothing and also any sort of reasonability. Under the assumption of reasonability, true nothingness is logically impossible. Which is pretty much the same thing as saying under the assumption that logic is real, nothingness is logically impossible. Of course, if logic isn’t real, than logically impossible or logically possible becomes moot.
And yes, that does imply that there must at least be existence, though it does not imply anything actually have that property. You can have no universe, but in order to have reason, you can’t have absolutely nothing - you must at least have a first principal, some “thing” whose nature of being is to be. Some starting point for reason and existence and all that.
This we shall call God.