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The Catholic Church has had forty-one antipopes through its history, an antipope being an uncanonically elected “Pope”. The Catholic Church has infallibly taught that a heretic cannot be a true Pope. For a Pope is a member of the Church and its visible head, but a heretic places himself outside the Church by his choice and thus ceases to be a member of the Church. Hence, a heretic cannot be a true Pope. The Code of Canon Law, Pope Paul IV, St. Francis de Sales, St. Alphonsus and St. Robert Bellarmine all take this same line of argument, and hence teach that a heretic cannot be a true Pope.
To wit, if a true Pope were to become a manifest heretic then he would cease to be the Pope. This teaching is rooted in the infallible dogma that a heretic is not a Catholic, and is not a member of the Church. Now when he [the Pope] is explicitly a heretic, writes St. Francis de Sales, he falls ipso facto from his dignity and out of the Church. St. Antoninus agrees: In the case in which the Pope would become a heretic, he would find himself by that fact alone and without any other sentence, separated from the Church. He could not be a heretic and remain Pope. Pope Paul IV the taught the same in his Cum ex apostolatus officio, that no Catholic may accept a person as a pope who can be shown to be a heretic. The Marian Apparition at Fátima on 13th May, the feast day of St. Robert Bellarmine, may have been an indication of Our Lady to consider the statements of St. Robert Bellarmine, especially those on the subject of heretical popes. Per St. Robert Bellarmine: A pope who is a manifest heretic automatically ceases to be Pope and head, just as he ceases automatically to be a Christian and a member of the Church. Wherefore, he can be judged and punished by the Church. This is the teaching of all the ancient Fathers who teach that manifest heretics immediately lose all jurisdiction. And elsewhere: …he who is not a Christian is not a member of the Church and a manifest heretic is not a Christian, as is clearly taught by St. Cyprian, St. Athanasius, St. Jerome and others; therefore the manifest heretic cannot be Pope.
Pope Pius XII, Mystici Corporis Christi
>For not every sin, however grave it may be, is such as of its own nature to sever a man from the Body of the Church, as does schism or heresy or apostasy.
Pope Innocent III, Eius exemplo
>By the heart we believe and by the mouth we confess the one Church, not of heretics, but the Holy Roman, Catholic, and Apostolic Church outside of which we believe that no one is saved.
And how do we judge that a person is a heretic St. Robert Bellarmine reasons: For men are not bound or able to read hearts, but when they see that someone is a heretic by his external works, they judge him to be a heretic pure and simple, and condemn him as a heretic.