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The Mitzvah of Tzitzis supercedes the prohibition of Kil'ayim. (1)
If a person dies without children and his only brother is a Mukeh Shechin (afflicted with boils), we do not force the widow to perform Yibum. (2)
A sorceress is punished with death by Sekilah. (3)
Although Rebbi Yehudah generally does not use the principle of Semuchim, in Mishneh Torah (Sefer Devarim) he does use
A man may marry a girl who had relations (out of wedlock) with his father or son. Rebbi Yehudah uses Semuchim to derive that it is forbidden to marry a girl who had relations (out of wedlock) with one's father.
Rebbi Yehudah uses Semuchim in Mishneh Torah when it is Muchach (there is compelling proof that the Torah put the two verses together for the purpose of teaching something) or it is Mufneh (there are extra words in the verse).
The prohibition of Sha'atnez applies only when one is wearing the garment with intent to benefit from the garment.
The prohibition of Sha'atnez applies even when one is not wearing the garment but merely draped the garment over him.
The prohibition of Sha'atnez applies only to the mixing of wool and linen, and not to any other materials. (4)
Techeles is dyed wool and thus it is Sha'atnez when placed on a linen garment. (5)
Rava says that strings of Tzitzis made of wool or linen may be placed on any garment. Strings made of any other material may be placed only on a garment that is the same material as the strings.
A BIT MORE
1. The Chachamim derive this through Semuchim: the proximity of the prohibition of Kil'ayim with the Mitzvah of Tzitzis teaches that one may wear Tzitzis that contains Kil'ayim.
2. Instead, we force him to perform Chalitzah.
3. Ben Azai derives through Semuchim that just as a woman who has relations with an animal is punished with Sekilah, so is a sorceress. Rebbi Yehudah derives this from the principle, "Anything that was included in the Klal and was separated from the Klal regarding a certain teaching, that teaching applies to the entire Klal." Accordingly, Ov and Yid'oni -- which were included in the Klal of sorceresses that are punished with death -- were separated from the Klal by means of giving them the punishment of Sekilah. Consequently, the punishment of Sekilah applies not only to Ov and Yid'oni but to all sorceresses that were included in the Klal.
4. Whenever the Torah uses the word "Begadim," it refers only to a garment of wool or linen.
5. If the Torah had not permitted wearing Tzitzis that is Sha'atnez, the Mitzvah of wearing Techeles would have applied only to a woolen garment, but not to a linen garment since it would have been Sha'atnez.
WEARING SHA'ATNEZ WITHOUT INTENT TO DERIVE BENEFIT
The Gemara derives from the verse, "Lo Silbash Sha'atnez," that the prohibition of Sha'atnez does not apply to a merchant who wears the garment merely for the purpose of displaying it to his customers. Rashi says that the merchant is not wearing the garment with the intent of deriving benefit from it, and therefore it is permitted to wear the garment even if it is Sha'atnez. The Tosfos Yeshanim asks that the Gemara is explaining the opinion of Rebbi Yehudah, and Rebbi Yehudah maintains that a prohibition applies even though there is no intent. Therefore, the Tosfos Yeshanim explains that the merchant is wearing the garment of Sha'atnez in such a way that he is not benefiting from the garment at all, and that is why it is permitted. However, if he would benefit from the garment in any way, it would be prohibited even if he has no intent to benefit from it.
HOW TO SEW AND SELL GARMENTS OF SHA'ATNEZ
Tailors may sew a garment of Sha'atnez in the normal manner as long as they do not intend to benefit from the garment, such as to receive protection from rain and sun. Merchants may sell Sha'atnez garments in the normal manner, as long as they do not intend to benefit from the garment to protect them from the rain and sun. The Tzenu'im (people inclined to be stringent) drape the garment on a pole. Some opinions even allow one to wear a Sha'atnez garment without intent to benefit, such as wearing a garment for the purpose of avoiding custom duties, or wearing it for the purpose of checking the size. (Shulchan Aruch YD 301:6)
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