Assume Hno: ¬ (∃B : set, B ∈ Fam ∧ ∃U : set, U ∈ Tx ∧ U ≠ Empty ∧ U ⊆ (closure_of X Tx B)).

Apply FalseE to the current goal.

We will prove False.

We prove the intermediate claim HBc: Baire_space_closed X Tx.

An exact proof term for the current goal is (Baire_space_imp_closed X Tx HB).

We prove the intermediate claim HTx: topology_on X Tx.

An exact proof term for the current goal is (andEL (topology_on X Tx) (∀Fam0 : set, countable_set Fam0 → (∀A : set, A ∈ Fam0 → closed_in X Tx A ∧ interior_of X Tx A = Empty) → interior_of X Tx (⋃ Fam0) = Empty) HBc).

We prove the intermediate claim HBc_prop: ∀Fam0 : set, countable_set Fam0 → (∀A : set, A ∈ Fam0 → closed_in X Tx A ∧ interior_of X Tx A = Empty) → interior_of X Tx (⋃ Fam0) = Empty.

An exact proof term for the current goal is (andER (topology_on X Tx) (∀Fam0 : set, countable_set Fam0 → (∀A : set, A ∈ Fam0 → closed_in X Tx A ∧ interior_of X Tx A = Empty) → interior_of X Tx (⋃ Fam0) = Empty) HBc).

Set FamC to be the term {closure_of X Tx B|B ∈ Fam}.

We prove the intermediate claim HFamC_count: countable_set FamC.

An exact proof term for the current goal is (countable_image Fam HcountFam (λB0 : set ⇒ closure_of X Tx B0)).

We prove the intermediate claim HclIntEmpty: ∀B : set, B ∈ Fam → interior_of X Tx (closure_of X Tx B) = Empty.

Let B be given.

Assume HBIn: B ∈ Fam.

Apply xm (interior_of X Tx (closure_of X Tx B) = Empty) to the current goal.

Assume H.

An exact proof term for the current goal is H.

Assume Hneq: interior_of X Tx (closure_of X Tx B) ≠ Empty.

We will prove interior_of X Tx (closure_of X Tx B) = Empty.

Apply FalseE to the current goal.

We will prove False.

Apply (nonempty_has_element (interior_of X Tx (closure_of X Tx B)) Hneq) to the current goal.

Let x be given.

Assume HxInt: x ∈ interior_of X Tx (closure_of X Tx B).

We prove the intermediate claim HexU: ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ⊆ closure_of X Tx B.

An exact proof term for the current goal is (SepE2 X (λx0 : set ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ closure_of X Tx B) x HxInt).

Apply HexU to the current goal.

Let U be given.

Assume HU: U ∈ Tx ∧ x ∈ U ∧ U ⊆ closure_of X Tx B.

We prove the intermediate claim HUcore: (U ∈ Tx ∧ x ∈ U) ∧ U ⊆ closure_of X Tx B.

An exact proof term for the current goal is HU.

We prove the intermediate claim HUinTx: U ∈ Tx ∧ x ∈ U.

An exact proof term for the current goal is (andEL (U ∈ Tx ∧ x ∈ U) (U ⊆ closure_of X Tx B) HUcore).

We prove the intermediate claim HUsub: U ⊆ closure_of X Tx B.

An exact proof term for the current goal is (andER (U ∈ Tx ∧ x ∈ U) (U ⊆ closure_of X Tx B) HUcore).

We prove the intermediate claim HUx: x ∈ U.

An exact proof term for the current goal is (andER (U ∈ Tx) (x ∈ U) HUinTx).

We prove the intermediate claim HUne: U ≠ Empty.

An exact proof term for the current goal is (elem_implies_nonempty U x HUx).

Apply Hno to the current goal.

We use B to witness the existential quantifier.

Apply andI to the current goal.

An exact proof term for the current goal is HBIn.

We use U to witness the existential quantifier.

Apply andI to the current goal.

We will prove U ∈ Tx ∧ U ≠ Empty.

Apply andI to the current goal.

An exact proof term for the current goal is (andEL (U ∈ Tx) (x ∈ U) HUinTx).

An exact proof term for the current goal is HUne.

An exact proof term for the current goal is HUsub.

We prove the intermediate claim HFamC_prop: ∀A : set, A ∈ FamC → closed_in X Tx A ∧ interior_of X Tx A = Empty.

Let A be given.

Assume HA: A ∈ FamC.

We prove the intermediate claim HexB: ∃B : set, B ∈ Fam ∧ A = closure_of X Tx B.

An exact proof term for the current goal is (ReplE Fam (λB0 : set ⇒ closure_of X Tx B0) A HA).

Apply HexB to the current goal.

Let B be given.

Assume HBpair: B ∈ Fam ∧ A = closure_of X Tx B.

We prove the intermediate claim HBIn: B ∈ Fam.

An exact proof term for the current goal is (andEL (B ∈ Fam) (A = closure_of X Tx B) HBpair).

We prove the intermediate claim HAeq: A = closure_of X Tx B.

An exact proof term for the current goal is (andER (B ∈ Fam) (A = closure_of X Tx B) HBpair).

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

Apply andI to the current goal.

We prove the intermediate claim Hcl: closure_of X Tx (closure_of X Tx B) = closure_of X Tx B ∧ closed_in X Tx (closure_of X Tx B).

An exact proof term for the current goal is (closure_idempotent_and_closed X Tx B HTx).

An exact proof term for the current goal is (andER (closure_of X Tx (closure_of X Tx B) = closure_of X Tx B) (closed_in X Tx (closure_of X Tx B)) Hcl).

An exact proof term for the current goal is (HclIntEmpty B HBIn).

We prove the intermediate claim HintUnion: interior_of X Tx (⋃ FamC) = Empty.

An exact proof term for the current goal is (HBc_prop FamC HFamC_count HFamC_prop).

We prove the intermediate claim HUnionEq: ⋃ FamC = X.

Apply set_ext to the current goal.

Let x be given.

Assume Hx: x ∈ ⋃ FamC.

We will prove x ∈ X.

Apply (UnionE_impred FamC x Hx) to the current goal.

Let A be given.

Assume HxA: x ∈ A.

Assume HA: A ∈ FamC.

We prove the intermediate claim HexB: ∃B : set, B ∈ Fam ∧ A = closure_of X Tx B.

An exact proof term for the current goal is (ReplE Fam (λB0 : set ⇒ closure_of X Tx B0) A HA).

Apply HexB to the current goal.

Let B be given.

Assume HBpair: B ∈ Fam ∧ A = closure_of X Tx B.

We prove the intermediate claim HAeq: A = closure_of X Tx B.

An exact proof term for the current goal is (andER (B ∈ Fam) (A = closure_of X Tx B) HBpair).

We prove the intermediate claim HclsubX: closure_of X Tx B ⊆ X.

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

We prove the intermediate claim Hxcl: x ∈ closure_of X Tx B.

rewrite the current goal using HAeq (from right to left) at position 1.

An exact proof term for the current goal is HxA.

An exact proof term for the current goal is (HclsubX x Hxcl).

Let x be given.

Assume HxX: x ∈ X.

We will prove x ∈ ⋃ FamC.

We prove the intermediate claim HxUF: x ∈ ⋃ Fam.

rewrite the current goal using HXeq (from right to left) at position 1.

An exact proof term for the current goal is HxX.

Apply (UnionE_impred Fam x HxUF) to the current goal.

Let B be given.

Assume HxB: x ∈ B.

Assume HBIn: B ∈ Fam.

We prove the intermediate claim HBsubX: B ⊆ X.

Let y be given.

Assume HyB: y ∈ B.

We will prove y ∈ X.

We prove the intermediate claim HyUF: y ∈ ⋃ Fam.

An exact proof term for the current goal is (UnionI Fam y B HyB HBIn).

rewrite the current goal using HXeq (from left to right) at position 1.

An exact proof term for the current goal is HyUF.

We prove the intermediate claim HclSup: B ⊆ closure_of X Tx B.

An exact proof term for the current goal is (subset_of_closure X Tx B HTx HBsubX).

We prove the intermediate claim Hxcl: x ∈ closure_of X Tx B.

An exact proof term for the current goal is (HclSup x HxB).

We prove the intermediate claim HclIn: closure_of X Tx B ∈ FamC.

An exact proof term for the current goal is (ReplI Fam (λB0 : set ⇒ closure_of X Tx B0) B HBIn).

An exact proof term for the current goal is (UnionI FamC x (closure_of X Tx B) Hxcl HclIn).

We prove the intermediate claim HintX: interior_of X Tx X = X.

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

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

rewrite the current goal using HUnionEq (from right to left) at position 2.

An exact proof term for the current goal is HintUnion.

We prove the intermediate claim HXE: X = Empty.

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

An exact proof term for the current goal is HintXEmpty.

An exact proof term for the current goal is (HXne HXE).