Proof from Mishneh that Hilchos Chol HaMoed is M'd'rabbanan

Question

If one looks into the Tur and Beis Yosef at the beginning of Hilchos Chol HaMoed in Simana 630 you will find a big maklokes rishonim if malacha on Chol HaMoed is an issur d'oraisa, d'rabbanan, or perhaps is d'oraisa however was given over to the Rabbis to determine what is ossur to do and what is muter to do on Chol HaMoed.

Someone wanted to bring a proof from a Mishnah in Chagigah that malacha on Chol HaMoed is only an d'rabbanan. It says in the Mishnah (Perek 1 Mishnah 8):

היתר נדרים פורחין באויר ואין להם על מה שיסמכו הלכות שבת חגיגות והמעילות הרי הם כהררים התלוין בשערה שהן מקרא מועט והלכות מרובות הדינין והעבודות הטהרות והטמאות ועריות יש להן על מה שיסמכו והן הן גופי תורה

In regards to the first statement that "Heter Nedarim" are "porchim b'avir" Rashi says that the ability of a Chacham to be matir a neder has a remez (hint) in the Torah, however doesn't have anything to "rely" on (lismoch) from the Torah (to say that it's m'doraisa. However there is a remez and the whole halacha was given over to the Chachamim in the Torah Shel b'Peh. This is a similar idea of what some Rishonim hold about Chol HaMoed that it is from the Torah however was given over to the Chachamim to decide what is muter or ossur. The person wanted to say that you see since the Mishnah didn't give hilchos Chol HaMoed as an example (pointing out that Mesectas Chagigah is in Seder Moed and would have been more fitting to give an example that is relavant to Seder Moed>) must be that the issue of issur malacha on Chol HaMoed is in fact only a d'rabbanan. As well that it's not included in the last group of halachos that are listed which do have a "smach" in the Torah. Since it's not in the first and not in the last, must be only d'rabbanan altogether.

My question is as follows:

This also idea seems rather difficult to me. Is it possible to bring such a proof from the Mishnah in such a way (specifically this idea and this Mishnah.) Is this way of learning from a Mishnah have any source or foundation?

Answer 1

As you your self postulate, you cannot bring a proof from a missing clause in a Mishna, as we have the rule of תנא ושייר - a Tana can leave out cases.

(Typically followed up with מאי שייר דהאי שייר שייר - what esle did you leave out, as there's always more than one clause left out, if somethig is left out.)

By the way: The Gemara in חגיגה on .דף יח seems to make it clear - from multiple Pesukim - that work on Chol HaMoed is d'Oraisa, yet Tosafos there claim that these are all Asmachtot. Hard to beleive that Tosafos would contradict a "proof from a Mishna" in the same Masechet.