I know you go by "stubborn", but please reread the quoted paragraph from Pius IX. If the Pope wanted to say your interpretation, he could have said so quite clearly. On the other hand, what he actually said is rather like a BOD formulation.
No YOU re-read it, but in context.
Right before your italicized text, the pope says what you are attempting to do, which is to believe that it is "possible to arrive at eternal salvation although living in error and alienated from the true faith and Catholic unity"
, which he literally says is a belief which
"is certainly opposed to Catholic teaching".
You likely missed that part, which is why I put emphasis on it with bold text thinking you couldn't miss it.
Further, immediately after your italicized text, the pope goes on to say:
8. "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, to whom "the custody of the vineyard has been committed by the Savior".
In order for you to understand it as you apparently do, you have to take the text you italicized completely and totally out of context. You have to completely ignore as if never written at all, everything that he says immediately before and immediately after the text you italicized. You managed to take what is talking about, namely, that there is no salvation outside of the Church, and turned it into an altogether meaningless formula.
Now the only question I have for you is, why do you do that rather than simply understanding it in the context he taught it?