DOTA 2 is in my library but it does not appear on the game list.

Please contact Valve. Nothing SG can do.

You have to play it.

had the same problem...

Q: I own Dota 2, why is it not filtered?
A: Our system can only detect if you own Dota 2 if you have played it. This is a limitation on Steam's side - unless it's shown on your game's list we won't be able to tell if you own it or not. To have it filtered you will have to install and play Dota 2 until it appears on your games list.

Allright, thanks guys.

Closing.