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To clarify, the RCC doesn’t claim that salvation is provided by the RCC.

On the contrary, from YOUR popes and Catechism.....

"Outside the Church there is no salvation"

846 How are we to understand this affirmation, often repeated by the Church Fathers?335 Re-formulated positively, it means that all salvation comes from Christ the Head through the Church which is his Body:

Basing itself on Scripture and Tradition, the Council teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and Baptism, and thereby affirmed at the same time the necessity of the Church which men enter through Baptism as through a door. Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it or to remain in it.336

Pope Leo XIII, Satis Cognitum (# 9): "The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium." Satis Cognitum (# 9): June 29, 1896:
http://www.vatican.va/holy_father/leo_xiii/encyclicals/documents/hf_l-xiii_enc_29061896_satis-cognitum_en.html

Pius 9, Quanto Conficiamur Moerore: “Also well known is the Catholic teaching that no one can be saved outside the Catholic Church. Eternal salvation cannot be obtained by those who oppose the authority and statements of the same Church and are stubbornly separated from the unity of the Church and also from the successor of Peter, the Roman Pontiff..”
-http://www.papalencyclicals.net/Pius09/p9quanto.htm

Pope Pius IX, Amantissimus: “There are other, almost countless, proofs drawn from the most trustworthy witnesses which clearly and openly testify with great faith, exactitude, respect and obedience that all who want to belong to the true and only Church of Christ must honor and obey this Apostolic See and Roman Pontiff." Pope Pius IX, Amantissimus (On The Care Of The Churches), Encyclical promulgated on April 8, 1862, # 3.
http://www.ewtn.com/library/ENCYC/P9AMANT2.HTM

Pope Pius IX (1846–1878), Encyclical Singulari Quidem March 17, 1856): “There is only one true, holy, Catholic Church, which is the Apostolic Roman Church. There is only one See founded on Peter by the word of the Lord, outside of which we cannot find either true faith or eternal salvation. He who does not have the Church for a mother cannot have God for a father, and whoever abandons the See of Peter on which the Church is established trusts falsely that he is in the Church. (On the Unity of the Catholic Church)
http://www.papalencyclicals.net/Pius09/p9singul.htm

Pope Pius XI, Mortalium Animos: Furthermore, in this one Church of Christ no man can be or remain who does not accept, recognize and obey the authority and supremacy of Peter and his legitimate successors. Did not the ancestors of those who are now entangled in the errors of Photius [the eastern “Orthodox” schismatics] and the reformers, obey the Bishop of Rome, the chief shepherd of souls?...Let none delude himself with obstinate wrangling. For life and salvation are here concerned...” Pope Pius XI, Mortalium Animos, PTC:873) The Promotion of True Religious Unity), 11, Encyclical promulgated on January 6, 1928, #11;
http://www.vatican.va/holy_father/pius_xi/encyclicals/documents/hf_p-xi_enc_19280106_mortalium-animos_en.html

Pius XII, Humani Generis (27,28): "Some say they are not bound by the doctrine, explained in Our Encyclical Letter of a few years ago, and based on the Sources of Revelation, which teaches that the Mystical Body of Christ and the Roman Catholic Church are one and the same thing.[6] Some reduce to a meaningless formula the necessity of belonging to the true Church in order to gain eternal salvation...These and like errors, it is clear, have crept in among certain of Our sons."
http://www.vatican.va/holy_father/pius_xii/encyclicals/documents/hf_p-xii_enc_12081950_humani-generis_en.html

Fourth Lateran Council (1215): "There is but one universal Church of the faithful, outside which no one at all is saved."

Fifth Lateran Council: Moreover, since subjection to the Roman pontiff is necessary for salvation for all Christ's faithful, as we are taught by the testimony of both sacred scripture and the holy fathers, and as is declared by the constitution of pope Boniface VIII of happy memory, also our predecessor, which begins Unam sanctam, we therefore...renew and give our approval to that constitution... Fifth Lateran CouncilSession 11, 19 December 1516,
http://www.piar.hu/councils/ecum18.htm

Pope Innocent III and Lateran Council IV: "One indeed is the universal Church of the faithful, outside which no one at all is saved, in which the priest himself is the sacrifice, Jesus Christ, whose body and blood are truly contained in the sacrament of the altar under the species of bread and wine; the bread (changed) into His body by the divine power of transubstantiation, and the wine into the blood, so that to accomplish the mystery of unity we ourselves receive from His (nature) what He Himself received from ours."

Pope Innocent III and Lateran Council IV (A.D. 1215) [considered infallible by some]

Therefore, if anyone says that it is not by the institution of Christ the lord himself (that is to say, by divine law) that blessed Peter should have perpetual successors in the primacy over the whole Church; or that the Roman Pontiff is not the successor of blessed Peter in this primacy: let him be anathema. — Vatican 1, Ses. 4, Cp. 1

St. Thomas Aquinas: It is also shown that to be subject to the Roman Pontiff is necessary for salvation. For Cyril says in his Thesaurus: “Therefore, brethren, if we imitate Christ so as to hear his voice remaining in the Church of Peter and so as not be puffed up by the wind of pride, lest perhaps because of our quarrelling the wily serpent drive us from paradise as once he did Eve.” And Maximus in the letter addressed to the Orientals [Greeks] says: “The Church united and established upon the rock of Peter’s confession we call according to the decree of the Savior the universal Church, wherein we must remain for the salvation of our souls and wherein loyal to his faith and confession we must obey him.” — St. Thomas Aquinas, Against the Errors of the Greeks, Pt. 2, ch. 36
http://dhspriory.org/thomas/ContraErrGraecorum.htm#b38

St. Frances Xavier Cabrini: "Many Protestants have almost the same practices as we, only they do not submit to the Holy Father and attach themselves to the true Ark of Salvation. They do not want to become Catholics and unite themselves under the banner of truth wherein alone there is true salvation. Of what avail is it, children, if Protestants lead naturally pure, honest lives, yet lack the Holy Ghost? They may well say: 'We do no harm; we lead good lives'; but, if they do not enter the true fold of Christ, all their protestations are in vain." St. Frances Xavier Cabrini, "Travels", Chicago: 1944, pp. 84, 71.

St. Ambrose, "Expl. of Luke: "The Lord severed the Jewish people from His kingdom, and heretics and schismatics are also severed from the kingdom of God and from the Church. Our Lord makes it perfectly clear that every assembly of heretics and schismatics belongs not to God, but to the unclean spirit." — St. Ambrose, "Expl. of Luke", ch.7, 91-95; PL 15; SS, vol. II, p. 85, (quoted in The Apostolic Digest, by Michael Malone, Book 4: "The Book of Christians", Chapter 2: "Those Who Reject Christ's Church are Anti-Christian").http://www.geocities.com/Athens/Troy/6480/catholics/apostolic4chp2.html

Pope Boniface VIII, Bull Unam sanctam (1302): "We are compelled in virtue of our faith to believe and maintain that there is only one holy Catholic Church, and that one is apostolic. This we firmly believe and profess without qualification. Outside this Church there is no salvation and no remission of sins, the Spouse in the Canticle proclaiming: 'One is my dove, my perfect one. One is she of her mother, the chosen of her that bore her' (Canticle of Canticles 6:8); which represents the one mystical body whose head is Christ, of Christ indeed, as God. And in this, 'one Lord, one faith, one baptism' (Ephesians 4:5). Certainly Noah had one ark at the time of the flood, prefiguring one Church which perfect to one cubit having one ruler and guide, namely Noah, outside of which we read all living things were destroyed… We declare, say, define, and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman Pontiff."

Pope Boniface VIII, Unam Sanctam (Promulgated November 18, 1302) "If, therefore, the Greeks or others say that they are not committed to Peter and to his successors, they necessarily say that they are not of the sheep of Christ, since the Lord says that there is only one fold and one shepherd (Jn.10:16). Whoever, therefore, resists this authority, resists the command of God Himself. " http://www.fordham.edu/halsall/source/b8-unam.html

Pope Eugene IV, Cantate Domino (1441): "The most Holy Roman Church firmly believes, professes and preaches that none of those existing outside the Catholic Church, not only pagans, but also Jews and heretics and schismatics, can have a share in life eternal; but that they will go into the "eternal fire which was prepared for the devil and his angels" (Matthew 25:41), unless before death they are joined with Her; and that so important is the unity of this ecclesiastical body that only those remaining within this unity can profit by the sacraments of the Church unto salvation, and they alone can receive an eternal recompense for their fasts, their almsgivings, their other works of Christian piety and the duties of a Christian soldier. No one, let his almsgiving be as great as it may, no one, even if he pour out his blood for the Name of Christ, can be saved, unless he remain within the bosom and the unity of the Catholic Church."

Pope Eugene IV and the Council of Florence: "The sacrosanct Roman Church...firmly believes, professes, and proclaims that..not only pagans, but also Jews and heretics and schismatics cannot become participants in eternal life but will depart `into everlasting fire...unless before the end of life the same have been added to the flock; and that..no one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has remained in the bosom and unity of the Catholic Church.”— Pope Eugene IV and the Council of Florence (Seventeenth Ecumenical Council), Cantate Domino, Bull promulgated on February 4, 1441 (Florentine style), [considered infallible by some]