I think I got it now, after studying parts of the next few pages of the book. It gets easier if we consider a couple of simpler problems first. First we prove
\{p,q\}\vdash p\land q
This is a special case of one of the theorems Kunen proves in the book. It's kind of funny that I couldn't figure out how to prove that p and q proves "p and q" without using the book.
The proof is based on the observation that p\rightarrow(q\rightarrow p\land q) is a tautology, which I see that you're already using in your pdf.
\begin{align*}1.\quad & p\hspace{8 cm} & \in\{p,q\}\\ 2.\quad & q & \in\{p,q\}\\ 3.\quad & p\rightarrow(q\rightarrow p\land q) & \text{tautology}\\ 4.\quad & q\rightarrow p\land q &\text{MP 1,3}\\5.\quad & p\land q & \text{MP 2,4}\end{align*}
Hm, the "align" environment behaves strangely here, but it looks good enough. Now consider the slightly more difficult problem
\{\forall x p(x),\forall x q(x)\}\vdash\forall x(p(x)\land q(x))
\begin{align*}1.\quad & \forall x p(x) & \in\{\forall x p(x),\forall x q(x)\}\\ 2.\quad & \forall x q(x) & \in\{\forall x p(x),\forall x q(x)\}\\ 3.\quad & \forall x( p(x)\rightarrow(q(x)\rightarrow p(x)\land q(x))) & \text{tautology}\\ 4.\quad & \forall x( p(x)\rightarrow(q(x)\rightarrow p(x)\land q(x))) \rightarrow (\forall x p(x)\rightarrow\forall x(q(x)\rightarrow p(x)\land q(x)) &\text{axiom 3}\\ 5.\quad & \forall x p(x)\rightarrow\forall x(q(x)\rightarrow p(x)\land q(x)) & \text{MP 3,4}\\ 6.\quad &\forall x(q(x)\rightarrow p(x)\land q(x)) & \text{MP 1,5}\\ 7.\quad & \forall x q(x)\rightarrow \forall x(p(x)\land q(x)) & \text{axiom 3}\\ 8.\quad & \forall x(p(x)\land q(x)) &\text{MP 2,7}<br />
\end{align*}
The claim we want to prove is
\emptyset\vdash\forall x(\forall y(x\in y\leftrightarrow x\in y)\ \land\ \forall y(y\in x\leftrightarrow y\in x))
Let's define p(x) and q(x) to be the \forall y(\text{blah-blah}) formulas above, so that what we want to prove takes the form
\emptyset\vdash\forall x(p(x)\land q(x))
With this notation, the proof is exactly identical to the proof of the second theorem above. Only the motivation for the first two lines will be different. This time, the text motivating the first two lines should say "tautology".