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Reminder this is a double-elimination bracket. You are only knocked out if you lose TWO matchups. You can follow along the progression of a double-elim bracket by looking at this picture.
If you lose a match, you drop to the "losers" bracket, where if you lose again, you are knocked out. The last remaining trainer in the losers bracket faces off against the last remaining trainer in the winners bracket. (Loser bracket leader needs to defeat winner bracket leader twice to knock them out).