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ffed4../f6827.. bday: 4110 doc published by PrGxv..
Param a842e.. : ι(ιι) → ι
Param 4a7ef.. : ι
Definition Subq := λ x0 x1 . ∀ x2 . prim1 x2 x0prim1 x2 x1
Known 3f849.. : ∀ x0 . Subq x0 4a7ef..x0 = 4a7ef..
Known 9b5af.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (a842e.. x0 x1)∀ x3 : ο . (∀ x4 . prim1 x4 x0prim1 x2 (x1 x4)x3)x3
Definition False := ∀ x0 : ο . x0
Known FalseE : False∀ x0 : ο . x0
Definition not := λ x0 : ο . x0False
Definition nIn := λ x0 x1 . not (prim1 x0 x1)
Known dcd83.. : ∀ x0 . nIn x0 4a7ef..
Theorem 5dcf0.. : ∀ x0 : ι → ι . a842e.. 4a7ef.. x0 = 4a7ef.. (proof)
Param 80242.. : ιο
Param and : οοο
Param 099f3.. : ιιο
Definition 02b90.. := λ x0 x1 . and (and (∀ x2 . prim1 x2 x080242.. x2) (∀ x2 . prim1 x2 x180242.. x2)) (∀ x2 . prim1 x2 x0∀ x3 . prim1 x3 x1099f3.. x2 x3)
Known and3I : ∀ x0 x1 x2 : ο . x0x1x2and (and x0 x1) x2
Theorem c0b78.. : ∀ x0 . (∀ x1 . prim1 x1 x080242.. x1)02b90.. x0 4a7ef.. (proof)
Theorem 80b57.. : ∀ x0 . (∀ x1 . prim1 x1 x080242.. x1)02b90.. 4a7ef.. x0 (proof)
Theorem 64de1.. : 02b90.. 4a7ef.. 4a7ef.. (proof)
Param 02a50.. : ιιι
Param e4431.. : ιι
Param 4ae4a.. : ιι
Param 0ac37.. : ιιι
Param SNoEq_ : ιιιο
Known 9ecaa.. : ∀ x0 x1 . 02b90.. x0 x1∀ x2 : ο . (80242.. (02a50.. x0 x1)prim1 (e4431.. (02a50.. x0 x1)) (4ae4a.. (0ac37.. (a842e.. x0 (λ x3 . 4ae4a.. (e4431.. x3))) (a842e.. x1 (λ x3 . 4ae4a.. (e4431.. x3)))))(∀ x3 . prim1 x3 x0099f3.. x3 (02a50.. x0 x1))(∀ x3 . prim1 x3 x1099f3.. (02a50.. x0 x1) x3)(∀ x3 . 80242.. x3(∀ x4 . prim1 x4 x0099f3.. x4 x3)(∀ x4 . prim1 x4 x1099f3.. x3 x4)and (Subq (e4431.. (02a50.. x0 x1)) (e4431.. x3)) (SNoEq_ (e4431.. (02a50.. x0 x1)) (02a50.. x0 x1) x3))x2)x2
Known 019ee.. : ∀ x0 . 0ac37.. 4a7ef.. x0 = x0
Known 2b8be.. : ∀ x0 x1 . 80242.. x080242.. x1e4431.. x0 = e4431.. x1SNoEq_ (e4431.. x0) x0 x1x0 = x1
Known ebb60.. : 80242.. 4a7ef..
Known ab02c.. : e4431.. 4a7ef.. = 4a7ef..
Param iff : οοο
Known SNoEq_I : ∀ x0 x1 x2 . (∀ x3 . prim1 x3 x0iff (prim1 x3 x1) (prim1 x3 x2))SNoEq_ x0 x1 x2
Known 2911f.. : ∀ x0 . prim1 x0 (4ae4a.. 4a7ef..)∀ x1 : ι → ο . x1 4a7ef..x1 x0
Theorem 4e105.. : 02a50.. 4a7ef.. 4a7ef.. = 4a7ef.. (proof)
Param 23e07.. : ιι
Known set_ext : ∀ x0 x1 . Subq x0 x1Subq x1 x0x0 = x1
Known cbec9.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (23e07.. x0)∀ x2 : ο . (80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x1 x0x2)x2
Known c8ed0.. : 80242.. (4ae4a.. 4a7ef..)
Param ordinal : ιο
Known aab4f.. : ∀ x0 . ordinal x0e4431.. x0 = x0
Param ba9d8.. : ιο
Known f3fb5.. : ∀ x0 . ba9d8.. x0ordinal x0
Known 3db81.. : ba9d8.. (4ae4a.. 4a7ef..)
Known f1083.. : prim1 4a7ef.. (4ae4a.. 4a7ef..)
Known f5194.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x1 x0prim1 x1 (23e07.. x0)
Known 44eea.. : ∀ x0 . ordinal x0∀ x1 . prim1 x1 x0099f3.. x1 x0
Theorem 47397.. : 23e07.. (4ae4a.. 4a7ef..) = 4ae4a.. 4a7ef.. (proof)
Param 5246e.. : ιι
Known 44ec0.. : ∀ x0 . ordinal x05246e.. x0 = 4a7ef..
Theorem f5a07.. : 5246e.. (4ae4a.. 4a7ef..) = 4a7ef.. (proof)
Param bc82c.. : ιιι
Known b71d0.. : ∀ x0 x1 . 80242.. x080242.. x180242.. (bc82c.. x0 x1)
Theorem 18470.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x280242.. (bc82c.. x0 (bc82c.. x1 x2)) (proof)
Param f4dc0.. : ιι
Known 706f7.. : ∀ x0 . 80242.. x080242.. (f4dc0.. x0)
Theorem 9cb52.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x280242.. (bc82c.. x0 (bc82c.. x1 (f4dc0.. x2))) (proof)
Known 58c5d.. : ∀ x0 x1 . 80242.. x080242.. x1f4dc0.. (bc82c.. x0 x1) = bc82c.. (f4dc0.. x0) (f4dc0.. x1)
Theorem c8dd9.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2f4dc0.. (bc82c.. x0 (bc82c.. x1 x2)) = bc82c.. (f4dc0.. x0) (bc82c.. (f4dc0.. x1) (f4dc0.. x2)) (proof)
Known 368c5.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. (bc82c.. x0 x1) x2
Known 8782d.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x0 x2099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1)
Theorem e1455.. : ∀ x0 x1 x2 x3 x4 . 80242.. x080242.. x180242.. x280242.. x380242.. x4099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3)099f3.. (bc82c.. x0 (bc82c.. x1 x4)) (bc82c.. x2 (bc82c.. x3 x4)) (proof)
Known 5144e.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. x2 (bc82c.. x0 x1)
Theorem db322.. : ∀ x0 x1 x2 x3 x4 . 80242.. x080242.. x180242.. x280242.. x380242.. x4099f3.. (bc82c.. x1 x0) (bc82c.. x2 x3)099f3.. (bc82c.. x0 (bc82c.. x4 x1)) (bc82c.. x2 (bc82c.. x3 x4)) (proof)
Known 08d14.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1)099f3.. x0 x2
Known 5c481.. : ∀ x0 . 80242.. x0bc82c.. (f4dc0.. x0) x0 = 4a7ef..
Known 97325.. : ∀ x0 . 80242.. x0bc82c.. x0 4a7ef.. = x0
Theorem b9ca8.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x0 (bc82c.. x2 x1)099f3.. (bc82c.. x0 (f4dc0.. x1)) x2 (proof)
Theorem 1dc86.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. (bc82c.. x2 x1) x0099f3.. x2 (bc82c.. x0 (f4dc0.. x1)) (proof)
Known f3bd7.. : ∀ x0 x1 . 80242.. x080242.. x1bc82c.. x0 x1 = bc82c.. x1 x0
Theorem 98ce3.. : ∀ x0 x1 x2 x3 x4 x5 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x5099f3.. (bc82c.. x0 (bc82c.. x1 x5)) (bc82c.. x3 (bc82c.. x4 x2))099f3.. (bc82c.. x0 (bc82c.. x1 (f4dc0.. x2))) (bc82c.. x3 (bc82c.. x4 (f4dc0.. x5))) (proof)
Param e6316.. : ιιι
Known ed5b9.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1∀ x2 : ο . (80242.. (e6316.. x0 x1)(∀ x3 . prim1 x3 (23e07.. x0)∀ x4 . prim1 x4 (23e07.. x1)099f3.. (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)) (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)))(∀ x3 . prim1 x3 (5246e.. x0)∀ x4 . prim1 x4 (5246e.. x1)099f3.. (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)) (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)))(∀ x3 . prim1 x3 (23e07.. x0)∀ x4 . prim1 x4 (5246e.. x1)099f3.. (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)) (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)))(∀ x3 . prim1 x3 (5246e.. x0)∀ x4 . prim1 x4 (23e07.. x1)099f3.. (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)) (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)))x2)x2
Theorem 278ed.. : ∀ x0 x1 . 80242.. x080242.. x180242.. (e6316.. x0 x1) (proof)
Theorem fa87a.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x380242.. (bc82c.. (e6316.. x2 x1) (bc82c.. (e6316.. x0 x3) (f4dc0.. (e6316.. x2 x3)))) (proof)
Known ca858.. : ∀ x0 x1 . 80242.. x080242.. x1∀ x2 : ο . (∀ x3 x4 . (∀ x5 . prim1 x5 x3∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 x4∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)e6316.. x0 x1 = 02a50.. x3 x4x2)x2
Known e76d1.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (5246e.. x0)∀ x2 : ο . (80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x0 x1x2)x2
Known c7cc7.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x0 x1099f3.. x1 x2099f3.. x0 x2
Theorem eb06e.. : ∀ x0 x1 . 80242.. x080242.. x1∀ x2 : ο . (∀ x3 x4 . 02b90.. x3 x4(∀ x5 . prim1 x5 x3∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 x4∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)e6316.. x0 x1 = 02a50.. x3 x4x2)x2 (proof)
Theorem b4f65.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . (∀ x8 . prim1 x8 x6∀ x9 : ο . (∀ x10 . prim1 x10 x2∀ x11 . prim1 x11 x3x8 = bc82c.. (e6316.. x10 x1) (bc82c.. (e6316.. x0 x11) (f4dc0.. (e6316.. x10 x11)))x9)(∀ x10 . prim1 x10 x4∀ x11 . prim1 x11 x5x8 = bc82c.. (e6316.. x10 x1) (bc82c.. (e6316.. x0 x11) (f4dc0.. (e6316.. x10 x11)))x9)x9)(∀ x8 . prim1 x8 x2∀ x9 . prim1 x9 x3prim1 (bc82c.. (e6316.. x8 x1) (bc82c.. (e6316.. x0 x9) (f4dc0.. (e6316.. x8 x9)))) x7)(∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x5prim1 (bc82c.. (e6316.. x8 x1) (bc82c.. (e6316.. x0 x9) (f4dc0.. (e6316.. x8 x9)))) x7)Subq x6 x7 (proof)
Known bf919.. : 5246e.. 4a7ef.. = 4a7ef..
Known 3e9e2.. : 23e07.. 4a7ef.. = 4a7ef..
Theorem b92db.. : ∀ x0 . 80242.. x0e6316.. x0 4a7ef.. = 4a7ef.. (proof)
Param 56ded.. : ιι
Known 5ccff.. : ∀ x0 : ι → ο . (∀ x1 . 80242.. x1(∀ x2 . prim1 x2 (56ded.. (e4431.. x1))x0 x2)x0 x1)∀ x1 . 80242.. x1x0 x1
Known f6a2d.. : ∀ x0 . 80242.. x0x0 = 02a50.. (23e07.. x0) (5246e.. x0)
Known 22361.. : ∀ x0 x1 x2 x3 . 02b90.. x0 x102b90.. x2 x3(∀ x4 . prim1 x4 x0099f3.. x4 (02a50.. x2 x3))(∀ x4 . prim1 x4 x1099f3.. (02a50.. x2 x3) x4)(∀ x4 . prim1 x4 x2099f3.. x4 (02a50.. x0 x1))(∀ x4 . prim1 x4 x3099f3.. (02a50.. x0 x1) x4)02a50.. x0 x1 = 02a50.. x2 x3
Known 23b01.. : ∀ x0 . 80242.. x002b90.. (23e07.. x0) (5246e.. x0)
Known 0888b.. : ∀ x0 x1 . 02b90.. x0 x1∀ x2 . prim1 x2 x0099f3.. x2 (02a50.. x0 x1)
Known b6795.. : f4dc0.. 4a7ef.. = 4a7ef..
Known 63df9.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (23e07.. x0)prim1 x1 (56ded.. (e4431.. x0))
Known 9c8cc.. : ∀ x0 x1 . 02b90.. x0 x1∀ x2 . prim1 x2 x1099f3.. (02a50.. x0 x1) x2
Known 54843.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (5246e.. x0)prim1 x1 (56ded.. (e4431.. x0))
Theorem 0d1f9.. : ∀ x0 . 80242.. x0e6316.. x0 (4ae4a.. 4a7ef..) = x0 (proof)
Known 9ec10.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . 80242.. x180242.. 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.. x180242.. x2x0 x1 x2
Known eb5ba.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. x1 (bc82c.. x0 x2)
Theorem 97a13.. : ∀ x0 x1 . 80242.. x080242.. x1e6316.. x0 x1 = e6316.. x1 x0 (proof)
Param 94f9e.. : ι(ιι) → ι
Known 696c0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0prim1 (x1 x2) (94f9e.. x0 x1)
Known 8948a.. : ∀ x0 . 80242.. x0f4dc0.. (f4dc0.. x0) = x0
Known 18a76.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x0 x1prim1 x1 (5246e.. x0)
Known 15454.. : ∀ x0 . 80242.. x0e4431.. (f4dc0.. x0) = e4431.. x0
Known 26c90.. : ∀ x0 x1 . 80242.. x080242.. x1099f3.. x0 (f4dc0.. x1)099f3.. x1 (f4dc0.. x0)
Known d4781.. : ∀ x0 x1 . 80242.. x080242.. x1099f3.. (f4dc0.. x0) x1099f3.. (f4dc0.. x1) x0
Known 8c6f6.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1)∀ x3 : ο . (∀ x4 . prim1 x4 x0x2 = x1 x4x3)x3
Known 4d4af.. : ∀ x0 x1 . 80242.. x080242.. x1099f3.. x0 x1099f3.. (f4dc0.. x1) (f4dc0.. x0)
Known 3cd4e.. : ∀ x0 x1 . 02b90.. x0 x1f4dc0.. (02a50.. x0 x1) = 02a50.. (94f9e.. x1 f4dc0..) (94f9e.. x0 f4dc0..)
Theorem 5fabd.. : ∀ x0 x1 . 80242.. x080242.. x1e6316.. (f4dc0.. x0) x1 = f4dc0.. (e6316.. x0 x1) (proof)

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