“… in cases involving mixed matters/mutual interests, the power of the Church prevails, because spiritual matters are more important than temporal … The contrary proposition is condemned in the Syllabus.”
This quote serves as a wonderful exemplar as to why laymen should make use of secondary sources rather than going to the primary source; you have completely misunderstood Pius IX’s Syllabus of Errors. It is doubtful you have given any thought as to why each thesis is contained in the contradictory proposition (“no opposition is so sharply determined as by the contradictory”), and so you proceed to assume, wrongly, that the negation of the proposition always holds true. Had you consulted with the Catholic Encyclopedia, for example, you would have realised your own error and also read:
“If, as for instance in thesis 42, the proposition, that in a conflict between civil and ecclesiastical laws the rights of the State should prevail, be condemned, then it does not follow from this thesis, that, in every conceivable case of conflicting laws the greater right is with the Church.”