How to get Mew?

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I found old Game Boy and games recently and among them there were Pokemon blue and Yellow. So I fired up the game, loaded my save file and saw I still have my 150 Pokemon, main 6 on maximum level. So I got them all, most of them I caught myself, others I traded. There used to be a rumor that you can get Mew, a mythical Pokemon number 151. I thought I'll encounter it in the cave after acquiring all 150 Pokemon and finishing the game but I never saw it. Is it just a myth that you can get Mew in the original games? what are the rules of finding it? Have you ever got it or even seen it?