The early tzars married into the family, just before Byzantium fell. Give me a second and will give you the quote.

The Russians, in the meantime, looked upon their own great city of Moscow as the “Third Rome.” And the fact that Russia was now the major Orthodox Christian power made the Russians feel that they were the direct inheritors of the Roman Empire. Russian leaders even titled themselves as “tsar,” which was the Roman equivalent of “Caesar.” It was during the reign of Russian Tsar Ivan III that the idea of Moscow being the third Rome would really crystalize. And the sentiment was most famously captured in an epistle written to him by one of his most faithful monks—Filofei of Pskov. Filofei wrote, “This present church of the third, new Rome, of thy sovereign Empire: The Holy Catholic Apostolic Church…shines in the whole universe more resplendent than the sun. And let it be known to the Lordship, oh pious Tsar, that all the empires of the Orthodox Christian faith have converged converged into thine one Empire. Thou art the sole Emperor of all the Christians in the whole universe. For two Romes have fallen, the third stands, and there shall be no fourth.” Yes, according to Filofei, Rome and Constantinople, two cities that claimed to be the center of Roman imperial might, had fallen. Yet Moscow still stood, and Filofei predicted it would remain standing since there would not be a “fourth” to rise up after it. Ivan III was the first Russian monarch to take up this mantle, calling himself a “tsar” and even marrying a Byzantine princess—Sophia Palaiologina—who was the niece of the last Byzantine emperor, Constantine XI.

This is in the intro of Pipes Russian empire.

https://www.amazon.com/gp/product/B09HF43ZNK/ref=ppx_yo_dt_b_d_asin_title_o00?ie=UTF8&psc=1