I'm trying to understand Kunen's Delta System Lemma proof (Set Theory: Introduction to Independence Proofs, Chapter II, Theorem 1.6). I'm heaving issues on understanding the last line:
"Since $|\alpha_0^{ < k}|<\theta$ there is an $r \subset \alpha_0$ and $B\subset A_2$ with $|B|=\theta$ and $\forall x \in B(x\cap \alpha_0=r)$, whence $B$ forms a $\Delta$-system with root $r$."
I didn't understand the whole sentence: Why is there such r and such B? And why is it a Delta system? I understood the previous parts of the proof, but I'm not understanding the ending :(
Adding more context: We want to construct a $\Delta$ system $B\subset A$ such that $|B|=0$. At this point, We already have constructed a subset $A_2$ of $A$ such that $|A_2|=\theta$ and $x \cap y\subset\alpha_0 < \theta $ whenever $x$ and $y$ are distinct members of $A_2$.
PS: A $\Delta$-system is a set $A$ such that there exists $r$ such that $x \cap y=r$ whenever $x$ and $y$ are distinct members of $A$.
At that point in the argument you know that $x\cap y\subseteq \alpha_0$ whenever $x$ and $y$ are distinct elements of $\mathscr{A}_2$. For each $x\in\mathscr{A}_2$ we know that $|x|<\kappa$, so $x\cap\alpha_0\in[\alpha_0]^{<\kappa}$. For each $z\in[\alpha_0]^{<\kappa}$ let $\mathscr{A}_2(z)=\{x\in\mathscr{A}_2:x\cap\alpha_0=z\}$. Then $$\mathscr{A}_2=\bigcup_{z\in[\alpha_0]^{<\kappa}}\mathscr{A}_2(z)\;.$$
$|\mathscr{A}_2|=\theta>|\alpha_0^{<\kappa}|$, and $\theta$ is regular, so $\theta$ is not the union of $|\alpha_0^{<\kappa}|$ properly smaller sets; thus, there must be some $r\in[\alpha_0]^{<\kappa}$ such that $|\mathscr{A}_2(r)|=\theta$.
Now take $\mathscr{B}=\mathscr{A}_2(r)$. For all $x\in\mathscr{B}$ we have $x\cap\alpha_0=r$, and $\{x\setminus\alpha_0:x\in\mathscr{B}\}$ is a pairwise disjoint family by the construction of $\mathscr{A}_2$, so $\mathscr{B}$ is a $\Delta$-system with root $r$.