The moon, as all other things, is born with your conception of it - and when you die the moon also dies (of course this death is only a conceptual death - it's the end of the idea of "I am" and with it of "moon").
It is the same for all things - they exist only in human minds - before conceptional thought there are no things at all.
This is quite obvious as without a thought process there simply is only this "direct experience" (another concept...) which is devoid of all thing-ness.
The idea that something will "still be there" when you are gone is built on the false premise that something - a multitude of separate things - is there in the first place. But all that is "there" are ideas of things, and of course these ideas are not only your ideas/concepts, but similar ideas are also entertained by a few billion other minds - and as such they "live on" in these minds... the concept of "moon" will as such survive the death of your conceptual "I am", but this is no proof of the moon existing in the first place.

