we know The Soul of the Church is the Holy Ghost.  It is not an invisible
extension of the mystical body which includes the unbaptized.
 
Pope Pius XII, Mystici Corporis, June 29, 1943:  “... Leo XIII, of immortal memory
in the Encyclical, “Divinum illud,” [expressed it] in these words: ‘Let it suffice to
state this, that, as Christ is the Head of the Church, the Holy Spirit is her
soul.’”464


we know the Church is essentially (i.e., in its essence) a Mystical Body.  
 
Pope Leo X, Fifth Lateran Council, Session 11, Dec. 19, 1516: “... the mystical
body, the Church (corpore mystico)...”465

Pope St. Pius X, Editae saepe (# 8), May 26, 1910: “... the Church, the Mystical
Body of Christ...”466
 
Pope Leo XII, Quod Hoc Ineunte (# 1), May 24, 1824: “... His mystical Body.”467


Therefore, to teach that one can be saved without belonging to the Body is to teach
that one can be saved without belonging to the Church, since the Church is a Body.
 
Can one can be saved by belonging to the Soul of the Church, while not belonging to her Body? I don't see how anyone who knows these facts can answer yes,


Pope Leo XIII, Satis Cognitum (# 3), June 29, 1896: “For this reason the Church is
so often called in Holy Writ a body, and even the body of Christ... From this it
follows that those who arbitrarily conjure up and picture to themselves a hidden
and invisible Church are in grievous and pernicious error... It is assuredly
impossible that the Church of Jesus Christ can be the one or the other, as that
man should be a body alone or a soul alone.  The connection and union of
both elements is as absolutely necessary to the true Church as the intimate
union of the soul and body is to human nature.  The Church is not something
dead: it is the body of Christ endowed with supernatural life.”

Had Pius XII not made those infallible quotes, all the BODers would still be teaching the "soul of the Church" theory, including Fenton. Look it up here on CI and see how many BODers still teach the soul of the Church theory. The majority of BODers still have not heard the news of Pius XII or do not want to accept it.

Look up how many times they quote the Catechism of Pius X quote as if it were infallible:

The Catechism of Pope St. Pius X, The Apostles’ Creed, “The Church in Particular,”
Q. 29: “Q. But if a man through no fault of his own is outside the Church, can he
be saved? A. If he is outside the Church through no fault of his, that is, if he is
in good faith, and if he has received Baptism, or at least has the implicit desire of
Baptism; and if, moreover, he sincerely seeks the truth and does God’s will as
best as he can, such a man is indeed separated from the body of the Church, but
is united to the soul of the Church and consequently is on the way of
salvation.”