Can we have external automorphisms over intersectional models?

Question

Is the following inconsistent:

By "intersectional" set I mean a set having the intersectional set of every nonempty subset of it, being an element of it.

$\forall S \subset M: S\neq \varnothing \to \bigcap S \in M$

Where: $\bigcap S = \{x \mid \forall s \in S \, (x \in s) \}$

Sets $(\varnothing, \{\varnothing\}, V_\omega, H_x, \operatorname {Fin}(\omega.2 \setminus \omega))$ are examples; where $H_x$ is the set of all sets hereditarily strictly subnumerous to $x$; and $\operatorname {Fin}(x)$ is the set of all finite subsets of $x$.

Let the ambient theory of models be $\sf ZF$-$\sf Reg.$. Can we have a model $M$ of say Mac Lane set theory (with or without Regularity, and without Choice) that admits a non-trivial external automorphism and at the same time have $M$ be an intersectional set?

Answer 1

One can easily make a model of ZF-Reg with numerous Quine atoms. Simply begin with a model of ZFCU, with numerous urelements, and then turn the urlements into Quine atoms, which are singleton sets $a=\{a\}$, and the result is a model of ZF-Reg. The atoms can be permuted, and these permutations extend to automorphisms of the whole universe.

But meanwhile, there can be supertransitive models of MacLane set theory or even ZFC-Reg inside the original model, with multiple Quine atoms. Such a model will be intersectional, but admit automorphisms.

For example, consider $M=V_{\omega_1}[A]$ in the original universe, with a set $A$ of urelements. In the new model, these turn into Quine atoms, and this model $M$ is supertransitive, hence intersectional, and a model of MacLane set theory (without Reg) and much more. But permutations of $A$ extend to automorphisms of $M$.