The SUCCESSORS to Peter and all the Apostles are still in full communion with each other. That fellow in Rome is not a successor to Peter, since his seat lost its succession when it fell away into schism. Antioch is as much a seat of Peter as Rome is, as well.
That is not true. All the early Church acknowledged that the Bishop of Rome was the successor of Peter.
The various Orthodox Churches are not in full communion with the successor of Peter.
This wasn't always the case:
Macedonius, Patriarch of Constantinople (466-516)
Macedonius declared, when desired by the Emperor Anastasius to condemn the Council of Chalcedon, that 'such a step without an Ecumenical Synod presided over by the Pope of Rome is impossible.' (Macedonius, Patr. Graec. 108: 360a (Theophan. Chronogr. pp. 234-346 seq.)
John VI, Patriarch of Constantinople (715)
The Pope of Rome, the head of the Christian priesthood, whom in Peter, the Lord commanded to confirm his brethren. (John VI, Epist. ad Constantin. Pap. ad. Combefis, Auctuar. Bibl. P.P. Graec.tom. ii. p. 211, seq.)
Sergius, Metropolitain of Cyprus (649)
Writing to Pope Theodore:
O Holy Head, Christ our God hath destined thy Apostolic See to be an immovable foundation and a pillar of the Faith. For thou art, as the Divine Word truly saith, Peter, and on thee as a foundation-stone have the pillars of the Church been fixed. (Sergius Ep. ad Theod. lecta in Sess. ii. Concil. Lat. anno 649)
St. Sophronius, Patriarch of Jerusalem (c. 638)
Teaching us all orthodoxy and destroying all heresy and driving it away from the God-protected halls of our holy Catholic Church. And together with these inspired syllables and characters, I accept all his (the pope's) letters and teachings as proceeding from the mouth of Peter the Coryphaeus, and I kiss them and salute them and embrace them with all my soul ... I recognize the latter as definitions of Peter and the former as those of Mark, and besides, all the heaven-taught teachings of all the chosen mystagogues of our Catholic Church. (Sophronius, Mansi, xi. 461)
Transverse quickly all the world from one end to the other until you come to the Apostolic See (Rome), where are the foundations of the orthodox doctrine. Make clearly known to the most holy personages of that throne the questions agitated among us. Cease not to pray and to beg them until their apostolic and Divine wisdom shall have pronounced the victorious judgement and destroyed from the foundation ...the new heresy. (Sophronius, [quoted by Bishop Stephen of Dora to Pope Martin I at the Lateran Council], Mansi, 893)
St. Theodore the Studite of Constantinople (759-826)
Writing to Pope Leo III:
Since to great Peter Christ our Lord gave the office of Chief Shepherd after entrusting him with the keys of the Kingdom of Heaven, to Peter or his successor must of necessity every novelty in the Catholic Church be referred. [Therefore], save us, oh most divine Head of Heads, Chief Shepherd of the Church of Heaven. (Theodore, Bk. I. Ep. 23)
Writing to Pope Paschal:
Hear, O Apostolic Head, divinely-appointed Shepherd of Christ's sheep, keybearer of the Kingdom of Heaven, Rock of the Faith upon whom the Catholic Church is built. For Peter art thou, who adornest and governest the Chair of Peter. Hither, then, from the West, imitator of Christ, arise and repel not for ever (Ps. xliii. 23). To thee spake Christ our Lord: 'And thou being one day converted, shalt strengthen thy brethren.' Behold the hour and the place. Help us, thou that art set by God for this. Stretch forth thy hand so far as thou canst. Thou hast strength with God, through being the first of all. (Letter of St. Theodore and four other Abbots to Pope Paschal, Bk. ii Ep. 12, Patr. Graec. 99, 1152-3)
Theodoret, Bishop of Cyrus in Syria (450)
A native of Antioch, Theodoret ruled under the Antiochean Patriarch.
I therefore beseech your holiness to persuade the most holy and blessed bishop (Pope Leo) to use his Apostolic power, and to order me to hasten to your Council. For that most holy throne (Rome) has the sovereignty over the churches throughout the universe on many grounds. (Theodoret, Tom. iv. Epist. cxvi. Renato, p. 1197).
If Paul, the herald of the truth, the trumpet of the Holy Spirit, hastened to the great Peter, to convey from him the solution to those in Antioch, who were at issue about living under the law, how much more do we, poor and humble, run to the Apostolic Throne (Rome) to receive from you (Pope Leo) healing for wounds of the the Churches. For it pertains to you to have primacy in all things; for your throne is adorned with many prerogatives. (Theodoret Ibid, Epistle Leoni)