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Accomplishment of the dex requirements
In order to move forward at the Mew quest, you need the owned data of all Kanto Pokémon excluded legendarys on your Pokédex. You also need at least 71 different evolution's in your Pokédex. Spawns for all Pokémon can be found at their specific pages.
More specific information regarding owned and evolved Pokémon and their mechanic in regards to the Dex can be found at the Pokédex page.
Articuno, Zapdos, Moltres and Mewtwo
The first step of this quest is to obtain the seen data of certain legendary Pokémon. The seen data of this Pokémon can be obtained by a battle. They appear as Pokémon sprited NPC's and are one time battles. Once you have fought them, the seen data is saved, no matter if you win or lose.
While the three legendary birds don't need specific requirements, besides having a Pokémon knowing HM03-Surf, Mewtwo can only be reached after doing the eponymous quest. Articuno is located at Seafoam B4F, Zapdos at Power Plant, Moltres at Victory Road and Mewtwo in Cerulean Cave B1F in Kanto. Moltres seen data should already be obtained in case Victory Road was already passed through.
Old Sea Map
Now, if all requirements are reached, speak to the Know it all Man in Celadon City. The Know it all Man, located in Celadon Office, will reward the Old Sea Map which is needed for the travel to Faraway island.
With the help of the Old Sea Map, it is possible to travel to Far Away Island with a Lapras located in the bay south of Pallet Town. The travel will only be successful when the interaction to Lapras is done between 12:00pm and 13:00pm Pokétime.
Once arrived, a small way follows to the second part of the island where Mew hides at a bush.
A interaction with the bush will trigger the fight with Mew. After the fight is over, you can leave the island with the sailor located at the beach on Faraway Island. In case Mew was not successfully caught, you have the chance to catch it again by beating Nikola.