The first millennium church recognized the primacy of three “Petrine Sees”: Rome, Alexandria and Antioch, in that order.
Each of
these sees can trace their apostolic succession to Saint Peter, as Pope Gregory I explains in this letter:
http://www.newadvent.org/fathers/360207040.htm
The key difference is that the jurisdiction of
Alexandria and Antioch was limited to their territory (Egypt and Syria) by the First Ecumenical Council, whereas no geographic limits were ever imposed on Rome.
Let the ancient customs in Egypt, Libya and Pentapolis prevail, that the Bishop of Alexandria have jurisdiction in all these, since the like is customary for the Bishop of Rome also. Likewise in Antioch and the other provinces, let the Churches retain their privileges.
Just a few points to add
Speaking of order, and how history sees it,
The ancient ranking of sees in order of importance, BC
before Constantinople 
is important to note.
- Rome
- Alexandria
- Antioch
- Jerusalem
after Constantinople
- Rome
- Constantinople
- Alexandria
- Antioch
- Jerusalem
Antioch and Alexandria being demoted like that behind Constantinople, shows they (those 2 cities) carried no weight as far as being “previous” sees of Peter. Rome OTOH, remained #1 because it was and continues to be the see of Peter.
Also,
while alive, a bishop’s see is where he is, NOT where he was. Peter’s last see is Rome,
and since that is where he died, that makes Rome the see in which succession to Peter exists. And historically, that is the way it has been seen.
Bishop Irenaeus ~180 a.d., 1 man away from the apostle John, lists the successors (12 bishops by name) to Peter in Rome, down to his day, in his work “Against Heresies” [Bk 3 ch 3 (
http://www.newadvent.org/fathers/0103303.htm) , paragraphs 2-3]
Popes ordain other bishops during their pontificate. That doesn’t make those bishops they ordain are then popes, nor successors to that pope .