Analysis of information sources in references of the Wikipedia article "使徒統緒" in Chinese language version.
Ignatius was the second successor to Peter and may actually have been consecrated by that Apostle or Saint Paul.
Because it would be too long in such a volume as this to enumerate the successions of all the churches, we point to the tradition of that very great and very ancient and universally known Church, which was founded and established at Rome, by the two most glorious Apostles, Peter and Paul: we point I say, to the tradition which this Church has from the Apostles, and to her faith proclaimed to men which comes down to our time through the succession of her bishops, and so we put to shame . . . all who assemble in unauthorized meetings. For with this Church, because of its superior authority, every Church must agree — that is the faithful everywhere — in communion with which Church the tradition of the Apostles has been always preserved by those who are everywhere
Ignatius was the second successor to Peter and may actually have been consecrated by that Apostle or Saint Paul.