If you mean how to consider yourself as beaten the kanto region in pokemon soulsilver, then you should find zapdos,moltres, and articuno. You should also beat Red (Ash Ketchum). Once you do that, you can catch Mewtwo.
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He disappears until you beat the Pokemon league again.
I think it's beat red
just beat red then go to cerulean city then go to the cave
Yes, after you beat the Pokemon league the second time, every time you beat the Pokemon league again red will reappear at the platform on the top of mount silver. But red's Pokemon will be a little stronger every time you re-battle him.
You can beat the Elite Four again, and battle Red again. If you beat lance and the elite four and Lance again then you are able to face Red at mt silver.