Tanna of the House of Elijah [is not Elijah the Prophet, but rather he is the Tanna mentioned in the introduction of the Rambam's Book of Seeds, Chapter 17. Yehoshua ben Perachiah, Nittai the Arbelite, Choni the Circle Maker, and Elijah. So it is also written in the Book of Be'er Sheva in a response and in the section (page 92a). (And in the Chain of Tradition (24b) it is written that he was in the days of Yehuda ben Tabai), and the eyes of Yavan encompass (Yuchsin in the glosses of the Rema to the Book of Foundation of the World) and in the chapter "Who Was" (94a) Tanna of the House of Elijah Rabbi Natan says: all the inhabitants under one star.
And in Pesachim (112a) Teddav'a, even though Rabbi Akiva said: make your Sabbath like a weekday, etc., and if according to the words of the Rambam that Teddav'a was ancient, how can it be said in the name of Rabbi Natan and Rabbi Akiva and in the name of Yehuda ben Teima (see there).
Furthermore, if Teddav'a was during the time of Rabbi Yehoshua ben Perachiah, which was three thousand and six hundred years before the creation of the world, the astonishment of the needy is great regarding what is stated about Teddav'a: six thousand years is the world, one thousand is void, one thousand is Torah, one thousand is the days of the Messiah, and due to our sins, many have gone out, for then it was before the beginning of the days of the Messiah, which is two thousand years]:
Seder_HaDorot,_Tanaim_and_Amoraim.468:1