Objection 1): The Gates of Hell cannot prevail against the Church, as Christ said (Matthew 16). He said He would be with His Church all days until the end of the world (Matthew 28). What you are saying is contrary to the promises of Christ.

Answer: No, indefectibility (the promise of Christ to always be with His Church, and that the gates of Hell will not prevail against it) means that the Church will, until the end of time, remain essentially what she is.  The indefectibility of the Church requires that at least a remnant of the Church will exist until the end of the world, and that a true pope will never authoritatively teach error to the entire Church.  It does not exclude antipopes posing as popes (as we’ve had numerous times in the past, even in Rome) or a counterfeit sect that reduces the adherents of the true Catholic Church to a remnant in the last days.  This is precisely what is predicted to occur in the last days and what happened during the Arian crisis. 

St. Athanasius: "Even if Catholics faithful to tradition are reduced to a handful, they are the ones who are the true Church of Jesus Christ."[1]

Further, it should be noted that the Church has defined that heretics are the gates of Hell which Our Lord mentioned in Matthew 16!

Pope Vigilius, Second Council of Constantinople, 553:
“… we bear in mind what was promised about the holy Church and Him who said the gates of Hell will not prevail against it (by these we understand the death-dealing tongues of heretics)…”[2]

Pope St. Leo IX, Sept. 2, 1053: “The holy Church built upon a rock, that is Christ, and upon Peter… because by the gates of Hell, that is, by the disputations of heretics which lead the vain to destruction, it would never be overcome.”[3]

St. Thomas Aquinas (+1262): “Wisdom may fill the hearts of the faithful, and put to silence the dread folly of heretics, fittingly referred to as the gates of Hell.”[4] (Intro. To Catena Aurea.)

Notice that heretics are the gates of Hell.  Heretics are not members of the Church.  That’s why a heretic could never be a pope.  The gates of Hell (heretics) could never have authority over the Church of Christ.  It’s not those who expose the heretical Vatican II antipopes who are asserting that the gates of Hell have prevailed against the Church; it’s those who obstinately defend them as popes, even though they can clearly be proven to be manifest heretics.

Pope Innocent III, Eius exemplo, Dec. 18, 1208:
“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.”[5]

St. Francis De Sales (17^th century), Doctor of the Church, The Catholic Controversy, pp. 305-306:  "Now when he [the Pope] is explicitly a heretic, he falls ipso facto from his dignity and out of the Church..."

There is not one teaching of the Catholic Church that can be quoted which is contrary to the fact that there is presently a counterfeit sect which has reduced the true Catholic Church to a remnant in the days of the Great Apostasy, which is presided over by antipopes who have falsely posed as popes.  Those who assert that the Vatican II sect is the Catholic Church assert that the Catholic Church officially endorses false religions and false doctrines.  This is impossible and would mean that the gates of Hell have prevailed against the Catholic Church.

See other Answers to the Most Common Objections Against Sedevacantism