There is a theorem on $\Delta$-system in Jech's set theory above; let us observe the last sentence in the proof. For every $\alpha$, how does it always hold? In other words, how is the $Z$ constructed? Any help will be appreciated. Thanks ahead:)
Am I right: Since $|\{X \in W: a \in X\}|< \omega_1$, for any $a$, hence for any $\xi<\alpha$, given $X_\xi$, we have $|St(\bigcup{X_\xi}, W)|<\omega_1$. Therefore we have some $X \in W\setminus St(\bigcup{X_\xi,W})$ such that $X_\alpha=X$, which disjoint from all $X_\xi$.