Following the link from the Catholic Encyclopedia to the Vatican Counsel in which the ex cathedra was affirmed as doctrine, one will find the following explanation:
“Faithfully adhering, therefore, to the tradition inherited from the beginning of the Christian Faith, we, with the approbation of the sacred council, for the glory of God our Saviour, for the exaltation of the Catholic religion, and the salvation of Christian peoples, teach and define, as a Divinely revealed dogma, that the Roman pontiff, when he speaks ex cathedra, that is, when he, in the exercise of his office as shepherd and teacher of all Christians, by virtue of his supreme Apostolic authority, decides that a doctrine concerning faith or morals is to be held by the entire Church he possesses, in consequence of the Divine aid promised him in St. Peter, that infallibility with which the Divine Saviour wished to have His Church furnished for the definition of doctrine concerning faith or morals; and that such definitions of the Roman pontiff are of themselves, and not in consequence of the Church’s consent, irreformable.”
Now bearing that in mind, let’s turn to the scripture that is always cited as being the basis for the line of popes and the authority of Rome
And I also say to you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not prevail against it.
5026. taute tow’-tay, and tauten tow’-tane, and tautes tow’-tace dative case, accusative case and genitive case respectively of the feminine singular of 3778; (towards or of) this:–her, + hereof, it, that, + thereby, the (same), this (same).
Last I checked, Peter was not a female. So what possibly, would “this” in that statement refer to? Let’s look back two verses.
Simon Peter answered and said, “You are the Christ, the Son of the living God.” Jesus answered and said to him, “Blessed are you, Simon Bar-Jonah, for flesh and blood has not revealed this to you, but My Father who is in heaven.
That which is being referred to and is feminine? Peter’s confession that Jesus is Christ, the Son of God. This is the stone upon which the entirety of the church has been built for without that fact, there is no church.