Let Fam be given.

Assume Hcount: countable_set Fam.

Assume Hclosed: ∀A : set, A ∈ Fam → closed_in X Tx A ∧ interior_of X Tx A = Empty.

We will prove interior_of X Tx (⋃ Fam) = Empty.

Set U to be the term {X ∖ A|A ∈ Fam}.

We prove the intermediate claim HUcount: countable_set U.

An exact proof term for the current goal is (countable_image Fam Hcount (λA0 : set ⇒ X ∖ A0)).

We prove the intermediate claim HUsub: U ⊆ Tx.

Let u be given.

Assume Hu: u ∈ U.

We prove the intermediate claim HexA: ∃A : set, A ∈ Fam ∧ u = X ∖ A.

An exact proof term for the current goal is (ReplE Fam (λA0 : set ⇒ X ∖ A0) u Hu).

Apply HexA to the current goal.

Let A be given.

Assume HA: A ∈ Fam ∧ u = X ∖ A.

We prove the intermediate claim HAin: A ∈ Fam.

An exact proof term for the current goal is (andEL (A ∈ Fam) (u = X ∖ A) HA).

We prove the intermediate claim Heq: u = X ∖ A.

An exact proof term for the current goal is (andER (A ∈ Fam) (u = X ∖ A) HA).

We prove the intermediate claim HAcl: closed_in X Tx A.

An exact proof term for the current goal is (andEL (closed_in X Tx A) (interior_of X Tx A = Empty) (Hclosed A HAin)).

We prove the intermediate claim Hopen: open_in X Tx (X ∖ A).

An exact proof term for the current goal is (open_of_closed_complement X Tx A HAcl).

We prove the intermediate claim HopenTx: (X ∖ A) ∈ Tx.

An exact proof term for the current goal is (open_in_elem X Tx (X ∖ A) Hopen).

rewrite the current goal using Heq (from left to right).

An exact proof term for the current goal is HopenTx.

We prove the intermediate claim HUprop: ∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx.

Let u be given.

Assume Hu: u ∈ U.

We prove the intermediate claim HuTx: u ∈ Tx.

An exact proof term for the current goal is (HUsub u Hu).

We prove the intermediate claim HexA: ∃A : set, A ∈ Fam ∧ u = X ∖ A.

An exact proof term for the current goal is (ReplE Fam (λA0 : set ⇒ X ∖ A0) u Hu).

Apply HexA to the current goal.

Let A be given.

Assume HA: A ∈ Fam ∧ u = X ∖ A.

We prove the intermediate claim HAin: A ∈ Fam.

An exact proof term for the current goal is (andEL (A ∈ Fam) (u = X ∖ A) HA).

We prove the intermediate claim Heq: u = X ∖ A.

An exact proof term for the current goal is (andER (A ∈ Fam) (u = X ∖ A) HA).

We prove the intermediate claim HintA: interior_of X Tx A = Empty.

An exact proof term for the current goal is (andER (closed_in X Tx A) (interior_of X Tx A = Empty) (Hclosed A HAin)).

We prove the intermediate claim Hdual: interior_of X Tx A = X ∖ closure_of X Tx (X ∖ A).

An exact proof term for the current goal is (interior_closure_complement_duality X Tx A HTx (closed_in_subset X Tx A (andEL (closed_in X Tx A) (interior_of X Tx A = Empty) (Hclosed A HAin)))).

We prove the intermediate claim HcomplEmpty: X ∖ closure_of X Tx (X ∖ A) = Empty.

rewrite the current goal using Hdual (from right to left).

An exact proof term for the current goal is HintA.

Apply andI to the current goal.

An exact proof term for the current goal is HuTx.

We prove the intermediate claim HcomplEmptyU: X ∖ closure_of X Tx u = Empty.

rewrite the current goal using Heq (from left to right).

An exact proof term for the current goal is HcomplEmpty.

Apply (set_ext (closure_of X Tx u) X) to the current goal.

An exact proof term for the current goal is (closure_in_space X Tx u HTx).

Let x be given.

Assume HxX: x ∈ X.

We will prove x ∈ closure_of X Tx u.

Apply (xm (x ∈ closure_of X Tx u)) to the current goal.

Assume H.

An exact proof term for the current goal is H.

Assume Hnot: x ∉ closure_of X Tx u.

Apply FalseE to the current goal.

We prove the intermediate claim Hxminus: x ∈ X ∖ closure_of X Tx u.

Apply setminusI to the current goal.

An exact proof term for the current goal is HxX.

An exact proof term for the current goal is Hnot.

We prove the intermediate claim HxE: x ∈ Empty.

rewrite the current goal using HcomplEmptyU (from right to left).

An exact proof term for the current goal is Hxminus.

An exact proof term for the current goal is (EmptyE x HxE).

We prove the intermediate claim HdenseInt: dense_in (intersection_over_family X U) X Tx.

An exact proof term for the current goal is ((andER (topology_on X Tx) (∀U0 : set, U0 ⊆ Tx → countable_set U0 → (∀u0 : set, u0 ∈ U0 → u0 ∈ Tx ∧ dense_in u0 X Tx) → dense_in (intersection_over_family X U0) X Tx) HB) U HUsub HUcount HUprop).

We prove the intermediate claim HsubFam: ∀A : set, A ∈ Fam → A ⊆ X.

Let A be given.

Assume HA: A ∈ Fam.

An exact proof term for the current goal is (closed_in_subset X Tx A (andEL (closed_in X Tx A) (interior_of X Tx A = Empty) (Hclosed A HA))).

We prove the intermediate claim HDeMorgan: X ∖ (intersection_over_family X U) = ⋃ Fam.

Apply (set_ext (X ∖ (intersection_over_family X U)) (⋃ Fam)) to the current goal.

Let x be given.

Assume Hx: x ∈ X ∖ (intersection_over_family X U).

We prove the intermediate claim HxX: x ∈ X.

An exact proof term for the current goal is (setminusE1 X (intersection_over_family X U) x Hx).

We prove the intermediate claim Hxnot: x ∉ intersection_over_family X U.

An exact proof term for the current goal is (setminusE2 X (intersection_over_family X U) x Hx).

We prove the intermediate claim HnotAll: ¬ (∀u0 : set, u0 ∈ U → x ∈ u0).

Assume Hall.

Apply Hxnot to the current goal.

We prove the intermediate claim HdefInt: intersection_over_family X U = {x0 ∈ X|∀u1 : set, u1 ∈ U → x0 ∈ u1}.

Use reflexivity.

rewrite the current goal using HdefInt (from left to right).

Apply (SepI X (λx0 : set ⇒ ∀u1 : set, u1 ∈ U → x0 ∈ u1) x HxX) to the current goal.

An exact proof term for the current goal is Hall.

We prove the intermediate claim Hexu: ∃u0 : set, ¬ (u0 ∈ U → x ∈ u0).

An exact proof term for the current goal is (not_all_ex_demorgan_i (λu0 : set ⇒ u0 ∈ U → x ∈ u0) HnotAll).

Apply Hexu to the current goal.

Let u0 be given.

Assume Hnimp: ¬ (u0 ∈ U → x ∈ u0).

We prove the intermediate claim Hu0: u0 ∈ U ∧ x ∉ u0.

An exact proof term for the current goal is (not_imp (u0 ∈ U) (x ∈ u0) Hnimp).

We prove the intermediate claim Hu0U: u0 ∈ U.

An exact proof term for the current goal is (andEL (u0 ∈ U) (x ∉ u0) Hu0).

We prove the intermediate claim Hxnotu0: x ∉ u0.

An exact proof term for the current goal is (andER (u0 ∈ U) (x ∉ u0) Hu0).

We prove the intermediate claim HexA: ∃A : set, A ∈ Fam ∧ u0 = X ∖ A.

An exact proof term for the current goal is (ReplE Fam (λA0 : set ⇒ X ∖ A0) u0 Hu0U).

Apply HexA to the current goal.

Let A be given.

Assume HA: A ∈ Fam ∧ u0 = X ∖ A.

We prove the intermediate claim HAin: A ∈ Fam.

An exact proof term for the current goal is (andEL (A ∈ Fam) (u0 = X ∖ A) HA).

We prove the intermediate claim Hu0eq: u0 = X ∖ A.

An exact proof term for the current goal is (andER (A ∈ Fam) (u0 = X ∖ A) HA).

We prove the intermediate claim HxA: x ∈ A.

Apply dneg to the current goal.

Assume HxnotA: x ∉ A.

Apply Hxnotu0 to the current goal.

rewrite the current goal using Hu0eq (from left to right).

Apply setminusI to the current goal.

An exact proof term for the current goal is HxX.

An exact proof term for the current goal is HxnotA.

An exact proof term for the current goal is (UnionI Fam x A HxA HAin).

Let x be given.

Assume Hx: x ∈ ⋃ Fam.

We will prove x ∈ X ∖ (intersection_over_family X U).

Apply (UnionE_impred Fam x Hx (x ∈ X ∖ (intersection_over_family X U))) to the current goal.

Let A be given.

Assume HxA: x ∈ A.

Assume HAin: A ∈ Fam.

We prove the intermediate claim HAsubX: A ⊆ X.

An exact proof term for the current goal is (HsubFam A HAin).

We prove the intermediate claim HxX: x ∈ X.

An exact proof term for the current goal is (HAsubX x HxA).

Apply setminusI to the current goal.

An exact proof term for the current goal is HxX.

Assume HxInt: x ∈ intersection_over_family X U.

We prove the intermediate claim Hall: ∀u0 : set, u0 ∈ U → x ∈ u0.

An exact proof term for the current goal is (SepE2 X (λx0 : set ⇒ ∀u0 : set, u0 ∈ U → x0 ∈ u0) x HxInt).

Set u0 to be the term X ∖ A.

We prove the intermediate claim Hu0U: u0 ∈ U.

An exact proof term for the current goal is (ReplI Fam (λA0 : set ⇒ X ∖ A0) A HAin).

We prove the intermediate claim Hxnu0: x ∉ u0.

Assume Hxu0: x ∈ u0.

An exact proof term for the current goal is (setminusE2 X A x Hxu0 HxA).

An exact proof term for the current goal is (Hxnu0 (Hall u0 Hu0U)).

We prove the intermediate claim HUnionU: ⋃ Fam = X ∖ (intersection_over_family X U).

Use symmetry.

An exact proof term for the current goal is HDeMorgan.

rewrite the current goal using HUnionU (from left to right).

Set I to be the term intersection_over_family X U.

We prove the intermediate claim HIsub: I ⊆ X.

Let z be given.

Assume Hz: z ∈ I.

An exact proof term for the current goal is (SepE1 X (λz0 : set ⇒ ∀u0 : set, u0 ∈ U → z0 ∈ u0) z Hz).

We prove the intermediate claim Hdual: interior_of X Tx (X ∖ I) = X ∖ closure_of X Tx (X ∖ (X ∖ I)).

An exact proof term for the current goal is (interior_closure_complement_duality X Tx (X ∖ I) HTx (setminus_Subq X I)).

rewrite the current goal using Hdual (from left to right).

rewrite the current goal using (setminus_setminus_eq X I HIsub) (from left to right).

We prove the intermediate claim HclEq: closure_of X Tx I = X.

An exact proof term for the current goal is HdenseInt.

rewrite the current goal using HclEq (from left to right).

We prove the intermediate claim Hxx: X ∖ X = Empty.

Apply Empty_Subq_eq to the current goal.

Let z be given.

Assume Hz: z ∈ X ∖ X.

We will prove z ∈ Empty.

We prove the intermediate claim HzX: z ∈ X.

An exact proof term for the current goal is (setminusE1 X X z Hz).

We prove the intermediate claim HznotX: z ∉ X.

An exact proof term for the current goal is (setminusE2 X X z Hz).

Apply FalseE to the current goal.

An exact proof term for the current goal is (HznotX HzX).

An exact proof term for the current goal is Hxx.