The exercise is the following (Kunen “Set Theory” 2011, Ex.2.9.10):
Work in $ZFC^-$ ($ZFC$ without the foundation axiom), and assume that there is an infinite set $T$ such that $x=\{x\}$ for all $x\in T$.
Let
$$M:=\{y\in WF(T) \mid\exists A \subset T \text{ finite, }\forall \pi \in Aut(T/A),\hat \pi (y) =y \}$$
(The definitions of $WF(T)$ etc are below).
Prove that $T\in M, M $is a transitive model of $ZF^-$,plus $(AC)^{WF}$ plus “$T$ cannot be well-ordered.”
$WF(T)$ is the class of well-founded sets starting from transitive set $T$:
・$R(0,T)=T$
・$R(\alpha +1,T)=P(R(\alpha,T))$
・$R(\gamma,T)= \bigcup_{\alpha\in \gamma}R(\alpha,T)$($\gamma$ is limit)
・$WF(T)= \bigcup_{\alpha\in ON}R(\alpha,T)$
$Aut(T/A)$ is like that of the field theory (in this condition now, any bijection on $T$ is automorphic about $\in$): $$Aut(T/A)=\{\pi :T\to T,\text{ bijection} \mid\pi\text{ on }A\text{ is the identity on }A\}$$
$\hat \pi$ is the unique extension of $\pi$ for $WF(T)$:
・$\hat \pi (x) =\pi (x) (x\in T)$
・$\hat \pi (x) =\{\hat\pi (z)\mid z\in x\}$(defined recursively)
I proved that $T\in M$ and $M$ is transitive, but I cannot completely prove the others. I cannot use the condition “$A$ is finite” very well to show that $M$ is the model of $ZF^-$ (especially Replacement axiom) $(AC)^{WF}$, plus “$T$ cannot be well-ordered.” Maybe AC will be used in the proof technically, but I have no idea...
I would appreciate any hints.
P.S. This is my first question on this site and I am a Japanese, so there may be some technical or English mistakes. Sorry.