A Proof of Zorn's Lemma

Question

I have a question about a step in following proof: A Simple Proof of Zorn's Lemma.pdf

Here the excerpt:

My question is why in $v \in P(B,y)$, Indeed we have $v < u <y$ by construction but why $v \in B$?

Answer 1

If $u \in P(B,y)$ then $u \in P(A,x)$, so if we have $v \in A$ and $v < u$ we have $v \in P(A,x)$. And we have $P(A,x) \subseteq B$, as he notes a couple of lines earlier, so $v \in B$ follows.