Proofgold Asset

Definition Subq := λ x0 x1 . ∀ x2 . prim1 x2 x0 ⟶ prim1 x2 x1
Param 1216a.. : ι → (ι → ο) → ι
Param 56ded.. : ι → ι
Param e4431.. : ι → ι
Param 099f3.. : ι → ι → ο
Definition 23e07.. := λ x0 . 1216a.. (56ded.. (e4431.. x0)) (λ x1 . 099f3.. x1 x0)
Known 572ea.. : ∀ x0 . ∀ x1 : ι → ο . Subq (1216a.. x0 x1) x0
Theorem e1ffe.. : ∀ x0 . Subq (23e07.. x0) (56ded.. (e4431.. x0)) (proof)
Definition 5246e.. := λ x0 . 1216a.. (56ded.. (e4431.. x0)) (099f3.. x0)
Theorem f4ff0.. : ∀ x0 . Subq (5246e.. x0) (56ded.. (e4431.. x0)) (proof)
Param 80242.. : ι → ο
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param bc82c.. : ι → ι → ι
Definition 02b90.. := λ x0 x1 . and (and (∀ x2 . prim1 x2 x0 ⟶ 80242.. x2) (∀ x2 . prim1 x2 x1 ⟶ 80242.. x2)) (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x1 ⟶ 099f3.. x2 x3)
Param 0ac37.. : ι → ι → ι
Param 94f9e.. : ι → (ι → ι) → ι
Known 9ec10.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . 80242.. x1 ⟶ 80242.. x2 ⟶ (∀ x3 . prim1 x3 (56ded.. (e4431.. x1)) ⟶ x0 x3 x2) ⟶ (∀ x3 . prim1 x3 (56ded.. (e4431.. x2)) ⟶ x0 x1 x3) ⟶ (∀ x3 . prim1 x3 (56ded.. (e4431.. x1)) ⟶ ∀ x4 . prim1 x4 (56ded.. (e4431.. x2)) ⟶ x0 x3 x4) ⟶ x0 x1 x2) ⟶ ∀ x1 x2 . 80242.. x1 ⟶ 80242.. x2 ⟶ x0 x1 x2
Definition TransSet := λ x0 . ∀ x1 . prim1 x1 x0 ⟶ Subq x1 x0
Definition ordinal := λ x0 . and (TransSet x0) (∀ x1 . prim1 x1 x0 ⟶ TransSet x1)
Param 02a50.. : ι → ι → ι
Known andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known 8a8cc.. : ∀ x0 . 80242.. x0 ⟶ ∀ x1 . 80242.. x1 ⟶ bc82c.. x0 x1 = 02a50.. (0ac37.. (94f9e.. (23e07.. x0) (λ x3 . bc82c.. x3 x1)) (94f9e.. (23e07.. x1) (bc82c.. x0))) (0ac37.. (94f9e.. (5246e.. x0) (λ x3 . bc82c.. x3 x1)) (94f9e.. (5246e.. x1) (bc82c.. x0)))
Known and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Known cbec9.. : ∀ x0 . 80242.. x0 ⟶ ∀ x1 . prim1 x1 (23e07.. x0) ⟶ ∀ x2 : ο . (80242.. x1 ⟶ prim1 (e4431.. x1) (e4431.. x0) ⟶ 099f3.. x1 x0 ⟶ x2) ⟶ x2
Known 0888b.. : ∀ x0 x1 . 02b90.. x0 x1 ⟶ ∀ x2 . prim1 x2 x0 ⟶ 099f3.. x2 (02a50.. x0 x1)
Known da962.. : ∀ x0 x1 x2 . prim1 x2 x0 ⟶ prim1 x2 (0ac37.. x0 x1)
Known 696c0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) (94f9e.. x0 x1)
Known e76d1.. : ∀ x0 . 80242.. x0 ⟶ ∀ x1 . prim1 x1 (5246e.. x0) ⟶ ∀ x2 : ο . (80242.. x1 ⟶ prim1 (e4431.. x1) (e4431.. x0) ⟶ 099f3.. x0 x1 ⟶ x2) ⟶ x2
Known 9c8cc.. : ∀ x0 x1 . 02b90.. x0 x1 ⟶ ∀ x2 . prim1 x2 x1 ⟶ 099f3.. (02a50.. x0 x1) x2
Known 287ca.. : ∀ x0 x1 x2 . prim1 x2 x1 ⟶ prim1 x2 (0ac37.. x0 x1)
Known 5a5d4.. : ∀ x0 x1 . 02b90.. x0 x1 ⟶ 80242.. (02a50.. x0 x1)
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Definition or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known edd11.. : ∀ x0 x1 x2 . prim1 x2 (0ac37.. x0 x1) ⟶ or (prim1 x2 x0) (prim1 x2 x1)
Known 8c6f6.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . prim1 x4 x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Param d3786.. : ι → ι → ι
Param SNoEq_ : ι → ι → ι → ο
Definition False := ∀ x0 : ο . x0
Definition not := λ x0 : ο . x0 ⟶ False
Definition nIn := λ x0 x1 . not (prim1 x0 x1)
Known 36ff9.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ 099f3.. x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . 80242.. x3 ⟶ prim1 (e4431.. x3) (d3786.. (e4431.. x0) (e4431.. x1)) ⟶ SNoEq_ (e4431.. x3) x3 x0 ⟶ SNoEq_ (e4431.. x3) x3 x1 ⟶ 099f3.. x0 x3 ⟶ 099f3.. x3 x1 ⟶ nIn (e4431.. x3) x0 ⟶ prim1 (e4431.. x3) x1 ⟶ x2) ⟶ (prim1 (e4431.. x0) (e4431.. x1) ⟶ SNoEq_ (e4431.. x0) x0 x1 ⟶ prim1 (e4431.. x0) x1 ⟶ x2) ⟶ (prim1 (e4431.. x1) (e4431.. x0) ⟶ SNoEq_ (e4431.. x1) x0 x1 ⟶ nIn (e4431.. x1) x0 ⟶ x2) ⟶ x2
Known 3eff2.. : ∀ x0 x1 x2 . prim1 x2 (d3786.. x0 x1) ⟶ and (prim1 x2 x0) (prim1 x2 x1)
Known c7cc7.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 099f3.. x0 x1 ⟶ 099f3.. x1 x2 ⟶ 099f3.. x0 x2
Known 18a76.. : ∀ x0 . 80242.. x0 ⟶ ∀ x1 . 80242.. x1 ⟶ prim1 (e4431.. x1) (e4431.. x0) ⟶ 099f3.. x0 x1 ⟶ prim1 x1 (5246e.. x0)
Known f5194.. : ∀ x0 . 80242.. x0 ⟶ ∀ x1 . 80242.. x1 ⟶ prim1 (e4431.. x1) (e4431.. x0) ⟶ 099f3.. x1 x0 ⟶ prim1 x1 (23e07.. x0)
Known b50ea.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ prim1 (e4431.. x0) (e4431.. x1) ⟶ prim1 x0 (56ded.. (e4431.. x1))
Known ordinal_TransSet : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known 4c9ee.. : ∀ x0 . 80242.. x0 ⟶ ordinal (e4431.. x0)
Theorem fb180.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ and (and (and (and (and (80242.. (bc82c.. x0 x1)) (∀ x2 . prim1 x2 (23e07.. x0) ⟶ 099f3.. (bc82c.. x2 x1) (bc82c.. x0 x1))) (∀ x2 . prim1 x2 (5246e.. x0) ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1))) (∀ x2 . prim1 x2 (23e07.. x1) ⟶ 099f3.. (bc82c.. x0 x2) (bc82c.. x0 x1))) (∀ x2 . prim1 x2 (5246e.. x1) ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x0 x2))) (02b90.. (0ac37.. (94f9e.. (23e07.. x0) (λ x2 . bc82c.. x2 x1)) (94f9e.. (23e07.. x1) (bc82c.. x0))) (0ac37.. (94f9e.. (5246e.. x0) (λ x2 . bc82c.. x2 x1)) (94f9e.. (5246e.. x1) (bc82c.. x0)))) (proof)
Theorem b71d0.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. (bc82c.. x0 x1) (proof)
Theorem 8782d.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 099f3.. x0 x2 ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1) (proof)
Param fe4bb.. : ι → ι → ο
Known 817af.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ fe4bb.. x0 x1 ⟶ or (099f3.. x0 x1) (x0 = x1)
Known 8dc73.. : ∀ x0 x1 . 099f3.. x0 x1 ⟶ fe4bb.. x0 x1
Known 4c8cc.. : ∀ x0 . fe4bb.. x0 x0
Theorem 742c4.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ fe4bb.. x0 x2 ⟶ fe4bb.. (bc82c.. x0 x1) (bc82c.. x2 x1) (proof)
Theorem 831d4.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 099f3.. x1 x2 ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x0 x2) (proof)
Theorem e46a2.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ fe4bb.. x1 x2 ⟶ fe4bb.. (bc82c.. x0 x1) (bc82c.. x0 x2) (proof)
Known c5dec.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 099f3.. x0 x1 ⟶ fe4bb.. x1 x2 ⟶ 099f3.. x0 x2
Theorem e418b.. : ∀ x0 x1 x2 x3 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ 099f3.. x0 x2 ⟶ fe4bb.. x1 x3 ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3) (proof)
Known 45d06.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ fe4bb.. x0 x1 ⟶ 099f3.. x1 x2 ⟶ 099f3.. x0 x2
Theorem f6ed7.. : ∀ x0 x1 x2 x3 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ fe4bb.. x0 x2 ⟶ 099f3.. x1 x3 ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3) (proof)
Theorem ba2a1.. : ∀ x0 x1 x2 x3 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ 099f3.. x0 x2 ⟶ 099f3.. x1 x3 ⟶ 099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3) (proof)
Known 9787a.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ fe4bb.. x0 x1 ⟶ fe4bb.. x1 x2 ⟶ fe4bb.. x0 x2
Theorem cc96f.. : ∀ x0 x1 x2 x3 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ fe4bb.. x0 x2 ⟶ fe4bb.. x1 x3 ⟶ fe4bb.. (bc82c.. x0 x1) (bc82c.. x2 x3) (proof)
Theorem fe60b.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ 02b90.. (0ac37.. (94f9e.. (23e07.. x0) (λ x2 . bc82c.. x2 x1)) (94f9e.. (23e07.. x1) (bc82c.. x0))) (0ac37.. (94f9e.. (5246e.. x0) (λ x2 . bc82c.. x2 x1)) (94f9e.. (5246e.. x1) (bc82c.. x0))) (proof)
Theorem 7a21c.. : ∀ x0 x1 x2 x3 . 02b90.. x0 x1 ⟶ 02b90.. x2 x3 ⟶ 02b90.. (0ac37.. (94f9e.. x0 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x2 (bc82c.. (02a50.. x0 x1)))) (0ac37.. (94f9e.. x1 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x3 (bc82c.. (02a50.. x0 x1)))) (proof)
Known 47e6b.. : ∀ x0 x1 . 0ac37.. x0 x1 = 0ac37.. x1 x0
Known aff96.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 94f9e.. x0 x1 = 94f9e.. x0 x2
Theorem f3bd7.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ bc82c.. x0 x1 = bc82c.. x1 x0 (proof)
Param 4a7ef.. : ι
Known 5ccff.. : ∀ x0 : ι → ο . (∀ x1 . 80242.. x1 ⟶ (∀ x2 . prim1 x2 (56ded.. (e4431.. x1)) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . 80242.. x1 ⟶ x0 x1
Known ebb60.. : 80242.. 4a7ef..
Known f6a2d.. : ∀ x0 . 80242.. x0 ⟶ x0 = 02a50.. (23e07.. x0) (5246e.. x0)
Known bf919.. : 5246e.. 4a7ef.. = 4a7ef..
Known d4dbc.. : ∀ x0 : ι → ι . 94f9e.. 4a7ef.. x0 = 4a7ef..
Known 019ee.. : ∀ x0 . 0ac37.. 4a7ef.. x0 = x0
Known set_ext : ∀ x0 x1 . Subq x0 x1 ⟶ Subq x1 x0 ⟶ x0 = x1
Known 3e9e2.. : 23e07.. 4a7ef.. = 4a7ef..
Theorem 85a07.. : ∀ x0 . 80242.. x0 ⟶ bc82c.. 4a7ef.. x0 = x0 (proof)
Theorem 97325.. : ∀ x0 . 80242.. x0 ⟶ bc82c.. x0 4a7ef.. = x0 (proof)
Param f4dc0.. : ι → ι
Known 2b8be.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ e4431.. x0 = e4431.. x1 ⟶ SNoEq_ (e4431.. x0) x0 x1 ⟶ x0 = x1
Known aab4f.. : ∀ x0 . ordinal x0 ⟶ e4431.. x0 = x0
Known 40f7a.. : ordinal 4a7ef..
Known e8942.. : ∀ x0 . Subq 4a7ef.. x0
Known c1313.. : ∀ x0 x1 . 02b90.. x0 x1 ⟶ ∀ x2 . 80242.. x2 ⟶ (∀ x3 . prim1 x3 x0 ⟶ 099f3.. x3 x2) ⟶ (∀ x3 . prim1 x3 x1 ⟶ 099f3.. x2 x3) ⟶ and (Subq (e4431.. (02a50.. x0 x1)) (e4431.. x2)) (SNoEq_ (e4431.. (02a50.. x0 x1)) (02a50.. x0 x1) x2)
Known 8948a.. : ∀ x0 . 80242.. x0 ⟶ f4dc0.. (f4dc0.. x0) = x0
Known 4d4af.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ 099f3.. x0 x1 ⟶ 099f3.. (f4dc0.. x1) (f4dc0.. x0)
Known 15454.. : ∀ x0 . 80242.. x0 ⟶ e4431.. (f4dc0.. x0) = e4431.. x0
Known 706f7.. : ∀ x0 . 80242.. x0 ⟶ 80242.. (f4dc0.. x0)
Theorem 5c481.. : ∀ x0 . 80242.. x0 ⟶ bc82c.. (f4dc0.. x0) x0 = 4a7ef.. (proof)
Theorem b5313.. : ∀ x0 . 80242.. x0 ⟶ bc82c.. x0 (f4dc0.. x0) = 4a7ef.. (proof)
Param 1beb9.. : ι → ι → ο
Known bfaa9.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . prim1 x1 (56ded.. x0) ⟶ ∀ x2 : ο . (prim1 (e4431.. x1) x0 ⟶ ordinal (e4431.. x1) ⟶ 80242.. x1 ⟶ 1beb9.. (e4431.. x1) x1 ⟶ x2) ⟶ x2
Known e3ccf.. : ∀ x0 . ordinal x0 ⟶ 80242.. x0
Known FalseE : False ⟶ ∀ x0 : ο . x0
Known dcd83.. : ∀ x0 . nIn x0 4a7ef..
Theorem ef972.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ 02b90.. (0ac37.. (94f9e.. (56ded.. x0) (λ x2 . bc82c.. x2 x1)) (94f9e.. (56ded.. x1) (bc82c.. x0))) 4a7ef.. (proof)
Known bc369.. : ∀ x0 . ordinal x0 ⟶ 23e07.. x0 = 56ded.. x0
Known 44ec0.. : ∀ x0 . ordinal x0 ⟶ 5246e.. x0 = 4a7ef..
Theorem c55f3.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ bc82c.. x0 x1 = 02a50.. (0ac37.. (94f9e.. (56ded.. x0) (λ x3 . bc82c.. x3 x1)) (94f9e.. (56ded.. x1) (bc82c.. x0))) 4a7ef.. (proof)
Known 32c48.. : ∀ x0 . 80242.. x0 ⟶ (∀ x1 . prim1 x1 (56ded.. (e4431.. x0)) ⟶ 099f3.. x1 x0) ⟶ ordinal x0
Known 41905.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ or (or (099f3.. x0 x1) (x0 = x1)) (099f3.. x1 x0)
Known In_irref : ∀ x0 . nIn x0 x0
Param 4ae4a.. : ι → ι
Param a842e.. : ι → (ι → ι) → ι
Known 9b3fd.. : ∀ x0 x1 . 02b90.. x0 x1 ⟶ and (and (and (and (80242.. (02a50.. x0 x1)) (prim1 (e4431.. (02a50.. x0 x1)) (4ae4a.. (0ac37.. (a842e.. x0 (λ x2 . 4ae4a.. (e4431.. x2))) (a842e.. x1 (λ x2 . 4ae4a.. (e4431.. x2))))))) (∀ x2 . prim1 x2 x0 ⟶ 099f3.. x2 (02a50.. x0 x1))) (∀ x2 . prim1 x2 x1 ⟶ 099f3.. (02a50.. x0 x1) x2)) (∀ x2 . 80242.. x2 ⟶ (∀ x3 . prim1 x3 x0 ⟶ 099f3.. x3 x2) ⟶ (∀ x3 . prim1 x3 x1 ⟶ 099f3.. x2 x3) ⟶ and (Subq (e4431.. (02a50.. x0 x1)) (e4431.. x2)) (SNoEq_ (e4431.. (02a50.. x0 x1)) (02a50.. x0 x1) x2))
Theorem c5221.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (bc82c.. x0 x1) (proof)
Known ordinal_ind : ∀ x0 : ι → ο . (∀ x1 . ordinal x1 ⟶ (∀ x2 . prim1 x2 x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . ordinal x1 ⟶ x0 x1
Known 4f62a.. : ∀ x0 x1 . ordinal x0 ⟶ prim1 x1 x0 ⟶ or (prim1 (4ae4a.. x1) x0) (x0 = 4ae4a.. x1)
Known 09af0.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . 80242.. x1 ⟶ prim1 (e4431.. x1) (4ae4a.. x0) ⟶ fe4bb.. x1 x0
Known 5faa3.. : ∀ x0 . prim1 x0 (4ae4a.. x0)
Known f1fea.. : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ Subq x0 x1 ⟶ fe4bb.. x0 x1
Known 958ef.. : ∀ x0 x1 . prim1 x1 (4ae4a.. x0) ⟶ or (prim1 x1 x0) (x1 = x0)
Known Subq_ref : ∀ x0 . Subq x0 x0
Known 44eea.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . prim1 x1 x0 ⟶ 099f3.. x1 x0
Known 09068.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . prim1 x1 x0 ⟶ prim1 (4ae4a.. x1) (4ae4a.. x0)
Known 5f6c1.. : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ 099f3.. x0 x1 ⟶ prim1 x0 x1
Known b72a3.. : ∀ x0 . ordinal x0 ⟶ ordinal (4ae4a.. x0)
Known ade9f.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . prim1 x2 x0 ⟶ 1beb9.. x2 x1 ⟶ prim1 x1 (56ded.. x0)
Known 5b4d8.. : ∀ x0 . ordinal x0 ⟶ 1beb9.. x0 x0
Theorem 327d7.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ bc82c.. (4ae4a.. x0) x1 = 4ae4a.. (bc82c.. x0 x1) (proof)
Theorem 9b91e.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ bc82c.. x0 (4ae4a.. x1) = 4ae4a.. (bc82c.. x0 x1) (proof)
Known ordinal_Hered : ∀ x0 . ordinal x0 ⟶ ∀ x1 . prim1 x1 x0 ⟶ ordinal x1
Theorem cbcc4.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ∀ x2 . prim1 x2 x0 ⟶ prim1 (bc82c.. x2 x1) (bc82c.. x0 x1) (proof)
Known dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known c47c0.. : ∀ x0 . not (099f3.. x0 x0)
Known 6ec49.. : ∀ x0 x1 x2 x3 . 02b90.. x0 x1 ⟶ 02b90.. x2 x3 ⟶ (∀ x4 . prim1 x4 x0 ⟶ 099f3.. x4 (02a50.. x2 x3)) ⟶ (∀ x4 . prim1 x4 x3 ⟶ 099f3.. (02a50.. x0 x1) x4) ⟶ fe4bb.. (02a50.. x0 x1) (02a50.. x2 x3)
Known 23b01.. : ∀ x0 . 80242.. x0 ⟶ 02b90.. (23e07.. x0) (5246e.. x0)
Known 027ee.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ or (099f3.. x0 x1) (fe4bb.. x1 x0)
Known orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known bba3d.. : ∀ x0 x1 . prim1 x0 x1 ⟶ nIn x1 x0
Theorem bf25c.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ ∀ x2 . prim1 x2 (23e07.. (bc82c.. x0 x1)) ⟶ or (∀ x3 : ο . (∀ x4 . and (prim1 x4 (23e07.. x0)) (fe4bb.. x2 (bc82c.. x4 x1)) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (prim1 x4 (23e07.. x1)) (fe4bb.. x2 (bc82c.. x0 x4)) ⟶ x3) ⟶ x3) (proof)
Theorem f9f27.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ ∀ x2 . prim1 x2 (5246e.. (bc82c.. x0 x1)) ⟶ or (∀ x3 : ο . (∀ x4 . and (prim1 x4 (5246e.. x0)) (fe4bb.. (bc82c.. x4 x1) x2) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (prim1 x4 (5246e.. x1)) (fe4bb.. (bc82c.. x0 x4) x2) ⟶ x3) ⟶ x3) (proof)
Known fbd1c.. : ∀ x0 : ι → ι → ι → ο . (∀ x1 x2 x3 . 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ x0 x4 x2 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x2)) ⟶ x0 x1 x4 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x3)) ⟶ x0 x1 x2 x4) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x2)) ⟶ x0 x4 x5 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x3)) ⟶ x0 x4 x2 x5) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x2)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x3)) ⟶ x0 x1 x4 x5) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x2)) ⟶ ∀ x6 . prim1 x6 (56ded.. (e4431.. x3)) ⟶ x0 x4 x5 x6) ⟶ x0 x1 x2 x3) ⟶ ∀ x1 x2 x3 . 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ x0 x1 x2 x3
Known 22361.. : ∀ x0 x1 x2 x3 . 02b90.. x0 x1 ⟶ 02b90.. x2 x3 ⟶ (∀ x4 . prim1 x4 x0 ⟶ 099f3.. x4 (02a50.. x2 x3)) ⟶ (∀ x4 . prim1 x4 x1 ⟶ 099f3.. (02a50.. x2 x3) x4) ⟶ (∀ x4 . prim1 x4 x2 ⟶ 099f3.. x4 (02a50.. x0 x1)) ⟶ (∀ x4 . prim1 x4 x3 ⟶ 099f3.. (02a50.. x0 x1) x4) ⟶ 02a50.. x0 x1 = 02a50.. x2 x3
Theorem 368c5.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ bc82c.. x0 (bc82c.. x1 x2) = bc82c.. (bc82c.. x0 x1) x2 (proof)
Theorem 47c39.. : ∀ x0 x1 x2 . 80242.. x0 ⟶ 80242.. x1 ⟶ 80242.. x2 ⟶ bc82c.. x0 x1 = bc82c.. x0 x2 ⟶ x1 = x2 (proof)
Theorem b6795.. : f4dc0.. 4a7ef.. = 4a7ef.. (proof)
Theorem 58c5d.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ f4dc0.. (bc82c.. x0 x1) = bc82c.. (f4dc0.. x0) (f4dc0.. x1) (proof)
Param If_i : ο → ι → ι → ι
Param e6316.. : ι → ι → ι
Definition 569d0.. := λ x0 x1 . If_i (x1 = 4a7ef..) 4a7ef.. (prim0 (λ x2 . and (80242.. x2) (e6316.. x2 x1 = x0)))
Param 236dc.. : ι → ι → ι
Param ce322.. : ι → ι
Param f6a32.. : ι → ι
Definition 8428d.. := λ x0 . 236dc.. (f4dc0.. (ce322.. x0)) (f4dc0.. (f6a32.. x0))
Param 1013b.. : ι → ο
Known 746bb.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ 1013b.. (236dc.. x0 x1)
Known 3a25a.. : ∀ x0 . 1013b.. x0 ⟶ 80242.. (ce322.. x0)
Known 43fc2.. : ∀ x0 . 1013b.. x0 ⟶ 80242.. (f6a32.. x0)
Theorem 0b595.. : ∀ x0 . 1013b.. x0 ⟶ 1013b.. (8428d.. x0) (proof)
Known a0d36.. : ∀ x0 . 236dc.. x0 4a7ef.. = x0
Known 55f9f.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ ce322.. (236dc.. x0 x1) = x0
Theorem 58212.. : ∀ x0 . 80242.. x0 ⟶ ce322.. x0 = x0 (proof)
Known 41b27.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ f6a32.. (236dc.. x0 x1) = x1
Theorem 551a1.. : ∀ x0 . 80242.. x0 ⟶ f6a32.. x0 = 4a7ef.. (proof)
Theorem 2b614.. : ce322.. 4a7ef.. = 4a7ef.. (proof)
Theorem fa6ce.. : f6a32.. 4a7ef.. = 4a7ef.. (proof)
Known c8ed0.. : 80242.. (4ae4a.. 4a7ef..)
Theorem 45e10.. : ce322.. (4ae4a.. 4a7ef..) = 4ae4a.. 4a7ef.. (proof)
Theorem 18d07.. : f6a32.. (4ae4a.. 4a7ef..) = 4a7ef.. (proof)
Definition f8050.. := 236dc.. 4a7ef.. (4ae4a.. 4a7ef..)
Theorem 76206.. : ce322.. f8050.. = 4a7ef.. (proof)
Theorem 5dab4.. : f6a32.. f8050.. = 4ae4a.. 4a7ef.. (proof)
Definition e37c0.. := λ x0 x1 . 236dc.. (bc82c.. (ce322.. x0) (ce322.. x1)) (bc82c.. (f6a32.. x0) (f6a32.. x1))
Theorem babfc.. : ∀ x0 x1 . 80242.. x0 ⟶ 80242.. x1 ⟶ bc82c.. x0 x1 = e37c0.. x0 x1 (proof)
Theorem 9fe90.. : ∀ x0 x1 . 1013b.. x0 ⟶ 1013b.. x1 ⟶ 1013b.. (e37c0.. x0 x1) (proof)
Known 501e2.. : ∀ x0 . 1013b.. x0 ⟶ x0 = 236dc.. (ce322.. x0) (f6a32.. x0)
Theorem 84397.. : ∀ x0 . 1013b.. x0 ⟶ e37c0.. 4a7ef.. x0 = x0 (proof)
Theorem 8e9bb.. : ∀ x0 . 1013b.. x0 ⟶ e37c0.. x0 4a7ef.. = x0 (proof)
Theorem 51295.. : ∀ x0 . 1013b.. x0 ⟶ e37c0.. (8428d.. x0) x0 = 4a7ef.. (proof)
Theorem eda89.. : ∀ x0 . 1013b.. x0 ⟶ e37c0.. x0 (8428d.. x0) = 4a7ef.. (proof)
Theorem 38cac.. : ∀ x0 . 80242.. x0 ⟶ f4dc0.. x0 = 8428d.. x0 (proof)
Param 7ce1c.. : ι → ι → ι
Definition df931.. := λ x0 x1 . If_i (x1 = 4a7ef..) 4a7ef.. (prim0 (λ x2 . and (1013b.. x2) (7ce1c.. x2 x1 = x0)))