This is not something that was claimed in 2002, 1952, 1932, 1432, 1032, 852, or even 432. This claim goes back to the very beginning right after the Apostles and the beginnings of the newly emerging Christian Church and comes directly from the singling out of the Apostle Peter by Christ and given ecclesiastical authority by Him.
It is thus written in Matthew: "Jesus replied, “Blessed are you, Simon son of Jonah! For this was not revealed to you by flesh and blood, but by My Father in heaven. 18
And I tell you that you are Peter, and on this rock I will build My church, and the gates of Hades will not prevail against it. 19I will give you the keys of the kingdom of heaven. Whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”…
Such a concept was accepted by the whole of Christendom. One Universal (Catholic) Christian Church - there was no other, with a leadership of Bishops with authority from God Incarnate which had been passed onto them.
Tertullian, considered on of the Early Church Fathers wrote in the 3rd century in his work "The Prescription Against the Heretics" the following: "For this is the way the apostolic Churches transmit their lists: like the Church of the Symrnaeans, which records that Polycarp was placed there by John; like the Church of the Romans, where Clement was ordained by Peter. In just this same way the other Churches display those whom they have as sprouts from the apostolic seed,
having been established in the episcopate by the Apostles."
And in 251 St. Cyprian of Carthage wrote : "And again He says to him [Peter] after His resurrection: 'Feed my sheep' (John 21:17). On him He builds the Church, and to him He gives the command to feed the sheep; and although He assigns a like power to all the Apostles,
yet He founded a single chair, and He established by His own authority a source and an intrinsic reason for that unity. Indeed, the others were that also which Peter was; but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. So too, all our shepherds, and the flock is shown to be one, fed by all the Apostles in single-minded accord. If someone does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he desert the chair of Peter upon whom the Church was built, can he still be confident that his is in the Church?"
There were no Baptists, no Episcopalians, no Mormons, no SDA, no Methodists, etc. Just one Universal (Catholic) Christian Church that over 95% (or more) of all Christians on the planet belonged to and which today's Roman Catholic Church is the direct descendant of.