I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks:
Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive.
Here we are working over the language $\mathcal{L} = \{\in\}$ of set theory, and $H(\aleph_1)$ is the set of hereditarily countable sets (i.e. those whose transitive closures are countable). The symbol $\preccurlyeq$ denotes "is an elementary submodel of".
Let's try the contrapositive: suppose $A$ is not transitive, so there are sets $u \in A$, $v \in u$ and $v \notin A$. I presume I want to find a formula $\varphi$ so that $A \vDash \varphi(u)$ but $H(\aleph_1) \nvDash \varphi(u)$. I am having trouble seeing how I can detect that $u$ contains an element $v$ which is not in $A$, since as far as $A$ is concerned, $v$ does not exist and $u$ behaves just like $u \cap A$.
If $u$ is a finite set, say of size 5, then if $\varphi(x)$ is the formula "$x$ has 5 distinct elements", we would have $H(\aleph_1) \vDash \varphi(u)$ but $A \vDash \neg \varphi(u)$. Something similar works if $u \cap A$ is finite. But if $u \cap A$ is infinite I don't see how to proceed.
Kunen gives the hint: "If $\emptyset \ne x \in A$, then there is an $f \in A$ such that $f :\omega \overset{\text{onto}}{\to} x$." I can't see how that helps. (It might be intended as a hint to the second part of the question, which asks for a countable transitive $A$ with a singleton subset that is not $\Delta_0$.)
Any hints are welcome.
HINT: (for Kunen's hint)
Let $A\prec H(\omega_1)$ and let $x\in A$.
First note that $\omega\in A$ and certainly $\omega\subseteq A$. Now use the hint, $x$ satisfies the formula "There exists a function $f$ whose domain is $\omega$ and $x$ is its range"; use elementarity to deduce that such $f$ exists in $A$, and that the "true range" (in $H(\omega_1)$, that is) of this $f$ is $x$. Conclude that $x\subseteq A$.