This definition is implausible because some descendants of Imperial Prince Koretada who were given the surname 'Minamoto no Ason' became the Koko-Genji; an example of this is MINAMOTO no Yasunao, a son of MINAMOTO no Yasuyuki, whose grandfather Prince Eiga was a son of Imperial Prince Koretada.
しかし、是忠親王の子孫でも源康尚（源康行の子。康尚の祖父・英我王は是忠親王の子）のように源朝臣を賜姓されて光孝源氏となった者もいる。