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Though believing in himself was part of what made Po prove himself as the Dragon Warrior, others like Tai Lung and the Furious Five were all shown to be confident in their chances of being chosen as well. And since the secret of the Dragon Scroll was meant to reveal who would be the Dragon Warrior, and the first person who came to understand its message was Po, it's apparent that he is the one and only Dragon Warrior.

The Dragon Warrior was also presumably selected only by Oogway, and "brought by the universe" as Oogway said. Oogway also spoke that whoever would be chosen would bring peace to the Valley of Peace and to Shifu, which also only applies to Po.