Mr. Simkin, the new math teacher at school, pulled aside three students, and handed them each a sealed envelope with a number written inside. He told them that they each had a positive integer, and the sum of their numbers was 14.

Sage and Rosemary opened their envelopes, and on seeing his number, Sage said, "I know that Rosemary and Tim have different numbers."

Rosemary replied, "I already knew our numbers were all different, and now I know everyone's number."

Tim said, "Then I do, as well," even though he never opened his envelope.

How could he have known? What numbers did everyone have?