They don't actually have to claim anything, the Yuan dynasty is widely considered by historians to be both Mongols and Chinese, especially after Kublai himself start becoming more Chinese, declared his territories a
Chinese dynasty, and took up the official tittle of "
Emperor of China" in additional to his "Great Khan" status. That pretty much makes the Yuan dynasty a Chinese dynasty with a Mongol emperor, with all the historic Mongol lands in its territories.
You can correctly say
Mongolia continued to be a part of China up until the Qing dynasty, when only the
outer half managed to
break free from Beijing, and no one in their right mind would be pissed at the stating of such dry-cut historical fact. (Beijing still have the
inner half of Mongolia in their posession, to the tune of 1,200,000 km2 of land.)