Hama the Bloodbender isn't exiling the cards she steals from opponents graveyard like I believe the card is supposed to?

Had a match on Avatar TLA premier draft and I played Hama, milled then exiled the opponents 'Airbending Lesson', which I then used on Hama itself with the waterbending ability. The next turn I replay Hama and I can still pick the same Airbending lesson card, I then repeat this 3 more times and I believe I could have done it as many times as I wanted if I had the mana. From how the card is written I think the card she takes is supposed to be fully exiled after use but I could be wrong.