## Axiomatisation of finite set theory just using membership

Require Import Undecidability.FOL.Util.Syntax.
Require Import Undecidability.FOL.Util.sig_bin.
Require Import Undecidability.FOL.Util.FullTarski.
Require Import Undecidability.FOL.Util.FullDeduction.
Import Vector.VectorNotations.
Require Import List.

Existing Instance falsity_on.

Notation term' := (term sig_func_empty).
Notation form' := (form sig_func_empty sig_pred_binary _ falsity_on).

Arguments Vector.nil {_}, _.
Arguments Vector.cons {_} _ {_} _, _ _ _ _.

Declare Scope syn'.
Open Scope syn'.

Notation "x ∈' y" := (atom sig_func_empty sig_pred_binary tt ([x; y])) (at level 35) : syn'.

Definition eq' (x y : term') :=
x`[] ∈' \$0 <~> y`[] ∈' \$0.

Notation "x ≡' y" := (eq' x y) (at level 35) : syn'.

Definition is_eset (t : term') :=
¬ (\$0 ∈' t`[]).

Definition is_adj (x y t : term') :=
\$0 ∈' t`[] <~> \$0 ∈' x`[] \$0 ≡' y`[].

Definition sub' (x y : term') :=
\$0 ∈' x`[] ~> \$0 ∈' y`[].

Definition ax_ext' :=
sub' \$1 \$0 ~> sub' \$0 \$1 ~> \$1 ≡' \$0.

Definition ax_eq_elem' :=
\$3 ≡' \$1 ~> \$2 ≡' \$0 ~> \$3 ∈' \$2 ~> \$1 ∈' \$0.

Definition ax_eset' :=
is_eset \$0.