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The prohibition and punishment of relations with a daughter (born out of wedlock) is derived from a Gezeirah Shavah.
The Tana of our Mishnah lists a daughter and granddaughters first, because they are a person's closest relatives. (1)
The Mishnah states that the 15 women are "Patur" instead of stating that they are "Asur" because the Tzaros of the 15 women are Asur only in a situation of a Mitzvah (when they are married to a brother), but not if they are married to a non-relative.
The Mishnah states that the 15 women are Patur from both Chalitzah and Yibum so that we should not think that they are Patur only from Yibum and not from Chalitzah. (2)
The Mishnah mentions Chalitzah prior to Yibum because the Tana of the Mishnah agrees with Aba Shaul who says that the Mitzvah of Chalitzah takes precedence over the Mitzvah of Yibum.
The Mishnah states a number in the Reisha and in the Seifa to exclude the opinion of Rav that a Tzaras Sotah is Patur, and to exclude the opinion of Rav Asi that a Tzaras Ailonis is Patur. (3)
According to Rav and R. Asi, the Mishnah does not include Tzaras Sotah and Tzaras Mema'enes because it is not possible for there to be a Tzaras Tzarah since she is forbidden to all of the brothers equally.
The Chachamim derive through a Gezeirah Shavah that the Mitzvah of Yibum does not apply to a wife's sister or to the Tzaros. All of the other 14 women are derived through a Mah Matzinu from the sister of a wife. (4)
If it was not for the Gezeirah Shavah that teaches that Yibum does not apply to a wife's sister, one would have thought that it does apply to her, even though she is prohibited, because of the concept of Aseh Docheh Lo Sa'aseh (a positive Mitzvah supersedes the prohibition of a negative Mitzvah.) (5)
A BIT MORE
1. Once the Tana was listing the man's children, he subsequently lists the woman's children. Once the Tana was listing her children, he follows with her mother and grandmothers. Once the Tana was listing her mother and grandmothers, he follows with his mother and grandmothers.
2. Chalitzah and Yibum are linked: anyone who is eligible for Yibum is also eligible for Chalitzah, and anyone who is not eligible for Yibum is also not eligible for Chalitzah.
3. If Rav and Rav Asi agree with each other's Dinim, the Mishnah would exclude the Tzarah of a girl who did Mi'un to the Yavam and the Tzarah of a divorcee who was remarried to the brother after she was married to someone else in the interim. If they do not agree with each other, the Mishnah would exclude the Din of the other and either the Tzarah of a girl who did Mi'un to the Yavam, or the Tzarah of a divorcee who was remarried.
4. The prohibitions against relations with all women which are forbidden with a Chiyuv Kares for Mezid and a Korban Chatas for Shogeg are derived from the sister of a wife, with the exception of the six women who are prohibited to all of the brothers and thus may never fall to Yibum in the first place.
5. The Gemara asks that since the 15 women are forbidden with a Chiyuv Kares, the concept of Aseh Docheh Lo Sa'aseh should not apply.
YIBUM IS ALWAYS A LO SA'ASEH THAT IS PUNISHED WITH KARES
Even though there is a concept of Aseh Docheh Lo Sa'aseh, the Gemara says that it should not apply to a Lo Sa'aseh that is punished with Kares. Tosfos asks that we should learn from the Mitzvah of Yibum itself, which allows the brother to marry his sister-in-law who is normally forbidden with a Chiyuv Kares, that the concept of Aseh Docheh Lo Sa'aseh does apply to a Lo Sa'aseh that is punished with Kares. Tosfos answers that the Mitzvah of Yibum is an exception and we cannot learn \other Mitzvos from it. The Mitzvah of Yibum applies only to a sister-in-law, and if the Mitzvah of Yibum did not supersede a Lo Sa'aseh that is punished with Kares, the Mitzvah would never apply at all. Other Mitzvos, however, should not supersede a Lo Sa'aseh that is punished with Kares, since the Mitzvah would still apply in other situations.
MARRYING THE TZARAH OF A FORBIDDEN WOMAN
If one of the 15 women is married to a non-relative and her husband dies, it is permitted to marry the Tzarah (the other wife who is not forbidden to him). (Shulchan Aruch EH 173:1)
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