Conditions for a valid pope

• male Catholic (not a heretic*)
• valid election

*See Pope Paul IV's bull (encyclical) Cum ex apostolatus officio on how a heretical papal candidate would be invalidly elected.

Who can judge the validity of a pope?

Only a valid Pope is the supreme judge:

Can. 1442 The Roman Pontiff is the supreme judge for the entire Catholic world; he renders judicial decisions personally, through the ordinary tribunals of the Apostolic See, or through judges he has delegated.

Antipope Anacletus II vs. Pope Innocent II

Antipope Anacletus II vs. Pope Innocent II is an illustrative example of a valid pope judging an invalid one. Pope Innocent was elected a mere 3 hours before Anacletus, and they both received episcopal consecration the same day. Anacletus, who had great clout from making his fortune off usury, drove Pope Innocent away from Rome; thus, Anacletus was considered the valid pope. When Anacletus died 8 years later, Pope Innocent II convened the Second Lateran Council, which annulled Anacletus's anti-papacy, reversed all the laws he enacted, and deposed the bishops and priests that Anacletus invalidly ordained.

Conciliarism

Even a council of all the world's bishops cannot depose a valid pope and elect a new one in his place. If they do, they elect an antipope. Every valid general or ecumenical council receives its full authority from a pope who ratifies its documents.

Those who think they can depose a valid pope believe in the heresy of conciliarism, which Fr. John Hardon, S.J., defines this way in his Modern Catholic Dictionary:

The theory that a general council of the Church is higher in authority than the Pope. It began in the fourteenth century, when respect for the papacy was undermined by confusion in Church and State. William of Ockham (1280-1349), in his battle with Pope John XXII (c. 1249-1334), questioned the divine institution of the primacy. Marsilius of Padua (1324) and John Jandun (1324) declared it was only a primacy of honor. During the great Western Schism (1378-1417) many otherwise reputable theologians, such as Peter of Ailly (1394) and John Gerson (1409) saw in the doctrine of the council's superiority over the Pope the only means of once more reuniting a divided Church. The viewpoint appeared that the Church in general was free from error, but the Church of Rome could err, and in fact had erred and fallen into heresy. The Council of Constance (1414-18), in its fourth and fifth sessions, declared for the superiority of council over Pope. However, these decisions never received papal approbation. In Gallicanism the conciliarist theory lived on for hundreds of years. Conciliarism was formally condemned by the First Vatican Council (1869-70), which defined papal primacy, declaring that the Pope had "full and supreme jurisdiction over the universal Church, not only in things which belong to faith and morals, but also in those which relate to the discipline and government of the Church spread throughout the world." He therefore possesses not merely the principal part but "all the fullness of this supreme power." Moreover, this power is ordinary or constant, and immediate or direct; it extends the Pope's authority over each and all the churches, whether local or territorial, and over each and all the churches, whether local or territorial, and over each and all the pastors and the faithful (Denzinger, 3063).

In more recent times, conciliarism has been renewed by those who appeal to a "magisterium of theologians" or "consensus of the people of God" against ordinary or even solemn teachings of the popes. (Etym. Latin concilium, council, assembly for consultation.)

Disagreeing who the valid pope is

Regarding the Great Western Schism,* when even saints disagreed about who the valid pope was,** Canon J. Didiot gives this analogy:

If after the election of a pope and before his death or resignation a new election takes place, it is null and schismatic; the one elected is not in the Apostolic Succession. This was seen at the beginning of what is called, somewhat incorrectly, the Great Schism of the West, which was only an apparent schism from a theological standpoint. If two elections take place simultaneously or nearly so, one according to laws previously passed and the other contrary to them, the apostolicity belongs to the pope legally chosen and not to the other, and though there be doubts, discussions, and cruel divisions on this point, as at the time of the so-called Western Schism, it is no less true, no less real that the apostolicity exists objectively in the true pope. What does it matter, in this objective relation, that it is not manifest to all and is not recognized by all till long after? A treasure is bequeathed to me, but I do not know whether it is in the chest A or in the casket B. Am I any less the possessor of this treasure?

^{*Technically, it was not a schism since all those involved were Catholic.
**From the "Western Schism" article: "St Catherine of Siena, St. Catherine of Sweden, Bl. Peter of Aragon, Bl. Ursulina of Parma, Philippe d'Alencon, and Gerard de Groote were in the camp of Urban; St. Vincent Ferrer, Bl. Peter of Luxemburg, and St. Colette belonged to the party of Clement."}

Any Catholic must refuse to follow an antipope.

However, any Catholic who recognizes an antipope as an antipope must refuse to follow him. St. Paul says twice in Galatian 1:8-9:

But although we, or an Angel from heaven, evangelize to you beside that which we have evangelized to you, be he anathema.

As we have said before, so now I say again, If any evangelize to you beside that which we have evangelized to you, be he anathema.

Notice, St. Paul says anyone, which certainly includes those who appear to be popes (antipopes).

Councils deposing antipopes

Anyone, including a council, can recognize that someone is an antipope, and a council can even elect a valid pope during the reigns of antipopes. That is not Conciliarism. Conciliarism is a council attempting to depose a valid pope.

"during the reign of an Apostate Pope, nobody can oppose him?"

Those who know he is an antipope must resist him.

"Does this not give credit to those that think a council has the power?"

Here's John of St. Thomas's opinion:

…[Deposition of an antipope] is to be made after a declaratory sentence about the crime…

…By the Church’s authority, a Council is able to be convened. That authority resides in the bishops, or in a majority of them. For the Church has the right from God to separate itself from a heretical popes [antipopes] and, consequently, to apply all the means that are in and of themselves necessary for such a separation. But it is a necessary means—and in and of itself—that such a crime be established juridically. Yet it cannot be established juridically unless a competent judgment be formed. In so grave a matter [of there being no valid pope], however, there cannot be a competent judgment except through a General Council. Since this matter deals with the universal head of the Church [that's not to say that an antipope is the head of the Church], it therefore pertains to the judgment of the Universal Church. That judgment is a General Council.

As a result, the principles–which sometimes maintain that the deposition of a Pontiff [antipope] belongs to God alone and at other times that he can be judged by inferiors in the case of heresy–come to be reconciled. Both are true. To God alone is reserved the casting out or deposition of a Pontiff (authoritatively and principally). Yet (ministerially and executionally) the Church makes judgment about a Pontiff [antipope] by declaring the crime and proposing that the [anti]pope should be avoided…

In response, a heretic must be avoided as a result of two rebukes that have been made juridically—by the Church’s authority and not according to private judgment. Great confusion would result in the Church if it would suffice that this rebuke should be made by a private individual. … Consequently, we see what the practice of the Church has been. When a[n anti]pope needed to be deposed, the case was treated first in a General Council before he was considered not to be a pope…

St. Jerome—in saying that a heretic departs on his own from the Body of Christ—does not preclude the Church’s judgment, especially in so grave a matter as is the deposition of a pope. He refers instead to the nature of that crime, which is such as to cut someone off from the Church on its own and without any other censure in addition to it–yet only so long as it should be declared by the Church. Although it separates from the Church on its own, the fact of separation does not [necessarily] make itself known to us [and for all Catholics] without that declaration.

This part is expressly contrary to St. Robert Bellarmine's view, in his De Romano Pontifice lib. II cap. 30, that a manifest heretical pope would become an antipope, even before deposition:

…So long as he has not become declared to us juridically as an infidel or heretic, be he ever so manifestly heretical according to private judgment, he remains as far as we are concerned a member of the Church, and consequently its head. Judgment is required from the Church, therefore, a judgment by which he would be proposed as not a Christian and to be avoided. It is only then that he ceases to be pope as far as we are concerned. As a result, however, he would not have ceased to be such even in and of himself, since all the things which he enacted had force in and of themselves.

^{Latin Original: Tractatus de Auctoritate Summi Pontificis, Disputatio II, Articulus III: Utrum Papa deponi possit ab Ecclesia, sicut ab eadem eligitur; et in quibus casibus? (Whether the Pope [antipope] may be deposed by the Church, in the same way as he is elected by the same (the Church), and in which cases?)}