Halakhah su Deuteronomio 22:11

Rav Kalischer (Drishat Tzion, Ma’amar Kadishin 3) argues that we could make bigdei kehunah without techeilet, just as, in the absence of techeilet, we wear tzitzit with white strings only (Menachot 38a). Most authorities reject his position, noting that the Tosefta (Menachot 6:6) explicitly states that bigdei kehunah cannot be made without techeilet. Moreover, the bigdei kehunah contain sha’atnez (a prohibited mixture of wool and linen; see Devarim 22:11). The positive commandment to make bigdei kehunah overrides this prohibition, but wearing bigdei kehunah that were made improperly would violate it.

And if one has a linen garment, he should place linen fringes in it, for the law is not like [the view of] Beit Shammai who taught (Menachot 40a) that a [linen] cloak is exempted from fringes. [This] means to say that since a linen garment could never have tekhelet placed on it, because [it would form] a forbidden mixture [of wool and linen] - as tekhelet must always be wool, since the tekhelet dye would never be absorbed properly by linen but rather only by wool, and the Torah obligates us in proper tekhelet - and since it would be impossible to ever put a string of wool into the linen because of the forbidden mixture, even white strings, meaning strings of linen, are not put on. This is what [they meant when] they said that Beit Shammai exempts a [linen] cloak from fringes. But the law is not like them, since their reason is because they did not expound adjacent verses; rather the law is like Beit Hillel who did expound adjacent verses. And they [therefore] held that there is no prohibition of forbidden mixtures in fringes, as the Torah stated (Deuteronomy 22:11-12), "Do not wear shaatnez (wool and linen together). Make fringes for yourself" - meaning to say, but fringes you may make for yourself from shaatnez. And therefore Beit Hillel held that a [linen] cloak is obligated in fringes, and the law is like them. Still, we do not put tekhelet on [a linen garment] due to a concern about indigo; that is, lest they would dye it with a non-tekhelet dye and then it would be shaatnez without the [performance of a] commandment. They asked in the Gemara in Menachot 40a, "Why? Is it not just white [strings]?" That is, for Beit Hillel who did expound adjacent verses, why should we make such a decree? As even if it is indigo, what is with that? Behold, we said that the forbidden mixture is permitted in fringes, meaning even with white [strings] - that is to say, even to put fringes of white wool in a garment of linen - and it is not [even] necessary to say that [it would be allowed] with fringes of wool that are dyed with tekhelet as is its commandment. And it was answered that these words that Beit Hillel said forbidden mixtures were permitted in fringes on the [strength of the] expounding of adjacent verses is only in cases where there is not from it; meaning to say when it is impossible from its own type - e.g. in a linen garment, if you want to place tekhelet in it, you will need wool regardless, as tekhelet is impossible without wool. [Only] in such cases did Beit Hillel rely on the expounding of adjacent verses and obligate the use of tekhelet in a linen garment. If however, there is 'from it,' meaning to say where you want to use white strings without tekhelet in a linen garment, it is not acceptable to use wool on the strength of the expounding of adjacent verses, as those white strings can be made of linen. This [follows] the principle of Reish Lakish; as Reish Lakish said that in any case where it is possible to maintain both the positive and negative commandments it is better [to do so], and only otherwise do we say that the positive commandment supersedes the negative. And here it is possible to maintain both commandments, for if you do not wish to place tekhelet with the white, linen suffices. And since this is so, if you use wool [you have violated the law of] forbidden mixtures and there is nothing [to help you]. Because there are many interpretations of these matters, I have spent some time on it for you. And from what I found in my trap I have set the table for you. But if you merit, my son, you will distinguish the truth. And if you perhaps find valid reason to challenge my words in this matter - or even in another - you should not mind the honor of your father and teacher, and I will then read (consider) your destruction as building.

To not wear shaatnez: To not wear clothing that is composed of wool and flax (linen), and that is what is called, shaatnez. And about this is it stated (Deuteronomy 22:11), "You shall not wear shaatnez, wool and flax together."