On a previous question that I asked, one correspondent pointed out the existence of a Mishnaic index, which lists every page of the early rabbinic literature on which there is a reference (or allusion) to each individual mishna. I would like to know if in the course of discussing individual mishnayot, the Bavli ends up referencing every single mishna altogether, or if some go totally unmentioned. Would a person who learns Mishna only through Shas Bavli still end up encountering the entire text?
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I misread the question and I wish to amend my answer. I did Shas with Daf Yomi during the 11th Cycle, and I did half of it again during the 12th. But I would never claim to have learned Shas Mishna based on that amount of study.
First, it is clear that there are no Bavli Tractates following many tractates of the Mishna. Only one tractate in the Seder of Zeraim, Brachos, has a Babylonian Gemara associated with it. That leaves Pe'ah, Demai, Kil'ayim, Shevi'is, Terumos, Ma'asros, Ma'aser Sheni, Challah, Orlah and Bikurim just from that seder! Mo'ed and Nashim are fully represented, I think in the Bavli. There are no Bavli gemaras for Avos and Eduyos in Seder Nezikin. From Seder Taharos, the Bavli is missing Keilim, Oholos, Negaim, Taharos, Mikvaos, Machshirin, Zavim, Yedayim, Tvul Yom and Uktzin.
Second, it may be true that many mishnayot are covered in discussions by reference, and one commentator has already pointed out that there is a major work that tries to track down all of the mishnayot cited therein. I have not reviewed that. But I have learned several of the mishnas listed above and there are some that come to mind, some in Challah for example, which I can't recall ever seeing in the Bavli.
Third, even if every mishna is somehow mentioned in the Bavli, I do not think that it is an appropriate way to learn the Mishnayos. They should be learned in the order that Rebbe placed them -- within context to each other.