Naming a fairy is notoriously difficult, because they don't particularly like to be named. There are so many of them, and humans have always wanted to categorize them. In that way, we think we can have more control over them.
Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.