I'm reading the section from Jech's Set Theory regarding the Lévy Collapse $Coll(\aleph_0,<\lambda)$. The following is a lemma towards proving the homogeneity of the Lévy Collapse:
Here Jech is using the boolean algebra version of forcing. In the second paragraph of the proof, he remarks that $B:C$ collapses $\check{\lambda}$ to $\check{\aleph_0}$ (where $B:C$ is the quotient of the boolean algebra $B$ by the filter generated by some generic ultrafilter over the boolean algebra $C$). I'm having trouble understanding why this holds without the initial assumption that $B$ collapses $\lambda $ to $\aleph_0$.
For instance, if $C$ is the trivial $0,1$ boolean algebra and $B$ is the power set algebra $P(\omega)$, then $B:C=B$ would be c.c.c. and thus not collapse $\lambda=|B|>\omega$ to $\omega$.
If this objection is correct, what is the correct proof of this lemma? Is there another resource that proves the homogeneity of the Lévy Collapse?