Some people say to nickname your starter Pokemon "Mew" and it will play as mew for the rest of the game but I never tested it...
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The only legitimate way to obtain Celebi and Mew is through a Nintendo event. In FireRed and LeafGreen you will need an event-only item to travel to the last two islands where you can catch those Pokemon
Mew was distributed at an event once, but that is long over and sadly if you want a mew in leafgreen now, you'll need help from other games or you could use cheat codes.
You cannot get Mew legit in Pokemon FireRed, for the event for Mew has ended. You would need to use a GameShark or trade for it from Pokemon Emerald.
How to battle mew in Fire Red. I don't know how??!!
There is none. all cheats for getting a mew are as fake as Micheal Jackson's nose