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 $\sf ZF$. Can we have a model $M$ of say Mac Lane set theory that admits an external automorphism and at the same time have $M$ be an intersectional set?

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?