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PUKhwZ4jaKa8zm9uNTqXaVnGmJWNTw5fEGT
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940d8../94db6.. bday: 36231 doc published by Pr4zB..
Param 4402e.. : ι(ιιο) → ο
Param cf2df.. : ι(ιιο) → ο
Definition SubqSubq := λ x0 x1 . ∀ x2 . x2x0x2x1
Param setminussetminus : ιιι
Param SingSing : ιι
Definition FalseFalse := ∀ x0 : ο . x0
Definition notnot := λ x0 : ο . x0False
Definition 8b6ad.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 . ∀ x5 : ο . ((x1 = x2∀ x6 : ο . x6)(x1 = x3∀ x6 : ο . x6)(x2 = x3∀ x6 : ο . x6)(x1 = x4∀ x6 : ο . x6)(x2 = x4∀ x6 : ο . x6)(x3 = x4∀ x6 : ο . x6)not (x0 x1 x2)not (x0 x1 x3)not (x0 x2 x3)not (x0 x1 x4)not (x0 x2 x4)not (x0 x3 x4)x5)x5
Definition c5756.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4(x1 = x5∀ x7 : ο . x7)(x2 = x5∀ x7 : ο . x7)(x3 = x5∀ x7 : ο . x7)(x4 = x5∀ x7 : ο . x7)not (x0 x1 x5)not (x0 x2 x5)x0 x3 x5x0 x4 x5x6)x6
Definition 2de86.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (c5756.. x0 x1 x2 x3 x4 x5(x1 = x6∀ x8 : ο . x8)(x2 = x6∀ x8 : ο . x8)(x3 = x6∀ x8 : ο . x8)(x4 = x6∀ x8 : ο . x8)(x5 = x6∀ x8 : ο . x8)not (x0 x1 x6)x0 x2 x6not (x0 x3 x6)x0 x4 x6not (x0 x5 x6)x7)x7
Definition 36d58.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (2de86.. x0 x1 x2 x3 x4 x5 x6(x1 = x7∀ x9 : ο . x9)(x2 = x7∀ x9 : ο . x9)(x3 = x7∀ x9 : ο . x9)(x4 = x7∀ x9 : ο . x9)(x5 = x7∀ x9 : ο . x9)(x6 = x7∀ x9 : ο . x9)x0 x1 x7not (x0 x2 x7)x0 x3 x7x0 x4 x7not (x0 x5 x7)not (x0 x6 x7)x8)x8
Definition af16d.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 . ∀ x9 : ο . (36d58.. x0 x1 x2 x3 x4 x5 x6 x7(x1 = x8∀ x10 : ο . x10)(x2 = x8∀ x10 : ο . x10)(x3 = x8∀ x10 : ο . x10)(x4 = x8∀ x10 : ο . x10)(x5 = x8∀ x10 : ο . x10)(x6 = x8∀ x10 : ο . x10)(x7 = x8∀ x10 : ο . x10)not (x0 x1 x8)not (x0 x2 x8)not (x0 x3 x8)x0 x4 x8not (x0 x5 x8)not (x0 x6 x8)not (x0 x7 x8)x9)x9
Definition a3794.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 . ∀ x10 : ο . (af16d.. x0 x1 x2 x3 x4 x5 x6 x7 x8(x1 = x9∀ x11 : ο . x11)(x2 = x9∀ x11 : ο . x11)(x3 = x9∀ x11 : ο . x11)(x4 = x9∀ x11 : ο . x11)(x5 = x9∀ x11 : ο . x11)(x6 = x9∀ x11 : ο . x11)(x7 = x9∀ x11 : ο . x11)(x8 = x9∀ x11 : ο . x11)x0 x1 x9not (x0 x2 x9)x0 x3 x9not (x0 x4 x9)not (x0 x5 x9)x0 x6 x9not (x0 x7 x9)x0 x8 x9x10)x10
Definition 6f07c.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 . ∀ x11 : ο . (a3794.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9(x1 = x10∀ x12 : ο . x12)(x2 = x10∀ x12 : ο . x12)(x3 = x10∀ x12 : ο . x12)(x4 = x10∀ x12 : ο . x12)(x5 = x10∀ x12 : ο . x12)(x6 = x10∀ x12 : ο . x12)(x7 = x10∀ x12 : ο . x12)(x8 = x10∀ x12 : ο . x12)(x9 = x10∀ x12 : ο . x12)x0 x1 x10x0 x2 x10not (x0 x3 x10)not (x0 x4 x10)x0 x5 x10not (x0 x6 x10)not (x0 x7 x10)x0 x8 x10not (x0 x9 x10)x11)x11
Definition 0e688.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ο . (6f07c.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10(x1 = x11∀ x13 : ο . x13)(x2 = x11∀ x13 : ο . x13)(x3 = x11∀ x13 : ο . x13)(x4 = x11∀ x13 : ο . x13)(x5 = x11∀ x13 : ο . x13)(x6 = x11∀ x13 : ο . x13)(x7 = x11∀ x13 : ο . x13)(x8 = x11∀ x13 : ο . x13)(x9 = x11∀ x13 : ο . x13)(x10 = x11∀ x13 : ο . x13)not (x0 x1 x11)x0 x2 x11not (x0 x3 x11)not (x0 x4 x11)not (x0 x5 x11)not (x0 x6 x11)not (x0 x7 x11)x0 x8 x11not (x0 x9 x11)not (x0 x10 x11)x12)x12
Definition f7297.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ο . (6f07c.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10(x1 = x11∀ x13 : ο . x13)(x2 = x11∀ x13 : ο . x13)(x3 = x11∀ x13 : ο . x13)(x4 = x11∀ x13 : ο . x13)(x5 = x11∀ x13 : ο . x13)(x6 = x11∀ x13 : ο . x13)(x7 = x11∀ x13 : ο . x13)(x8 = x11∀ x13 : ο . x13)(x9 = x11∀ x13 : ο . x13)(x10 = x11∀ x13 : ο . x13)x0 x1 x11not (x0 x2 x11)not (x0 x3 x11)not (x0 x4 x11)x0 x5 x11not (x0 x6 x11)not (x0 x7 x11)not (x0 x8 x11)not (x0 x9 x11)not (x0 x10 x11)x12)x12
Definition f3db1.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . ∀ x13 : ο . (f7297.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11(x1 = x12∀ x14 : ο . x14)(x2 = x12∀ x14 : ο . x14)(x3 = x12∀ x14 : ο . x14)(x4 = x12∀ x14 : ο . x14)(x5 = x12∀ x14 : ο . x14)(x6 = x12∀ x14 : ο . x14)(x7 = x12∀ x14 : ο . x14)(x8 = x12∀ x14 : ο . x14)(x9 = x12∀ x14 : ο . x14)(x10 = x12∀ x14 : ο . x14)(x11 = x12∀ x14 : ο . x14)not (x0 x1 x12)x0 x2 x12not (x0 x3 x12)not (x0 x4 x12)not (x0 x5 x12)not (x0 x6 x12)not (x0 x7 x12)x0 x8 x12not (x0 x9 x12)not (x0 x10 x12)x0 x11 x12x13)x13
Definition 02f3e.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (2de86.. x0 x1 x2 x3 x4 x5 x6(x1 = x7∀ x9 : ο . x9)(x2 = x7∀ x9 : ο . x9)(x3 = x7∀ x9 : ο . x9)(x4 = x7∀ x9 : ο . x9)(x5 = x7∀ x9 : ο . x9)(x6 = x7∀ x9 : ο . x9)x0 x1 x7x0 x2 x7x0 x3 x7x0 x4 x7not (x0 x5 x7)not (x0 x6 x7)x8)x8
Definition c8b10.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 . ∀ x9 : ο . (02f3e.. x0 x1 x2 x3 x4 x5 x6 x7(x1 = x8∀ x10 : ο . x10)(x2 = x8∀ x10 : ο . x10)(x3 = x8∀ x10 : ο . x10)(x4 = x8∀ x10 : ο . x10)(x5 = x8∀ x10 : ο . x10)(x6 = x8∀ x10 : ο . x10)(x7 = x8∀ x10 : ο . x10)x0 x1 x8not (x0 x2 x8)not (x0 x3 x8)x0 x4 x8not (x0 x5 x8)not (x0 x6 x8)not (x0 x7 x8)x9)x9
Definition 7db3a.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 . ∀ x10 : ο . (c8b10.. x0 x1 x2 x3 x4 x5 x6 x7 x8(x1 = x9∀ x11 : ο . x11)(x2 = x9∀ x11 : ο . x11)(x3 = x9∀ x11 : ο . x11)(x4 = x9∀ x11 : ο . x11)(x5 = x9∀ x11 : ο . x11)(x6 = x9∀ x11 : ο . x11)(x7 = x9∀ x11 : ο . x11)(x8 = x9∀ x11 : ο . x11)not (x0 x1 x9)not (x0 x2 x9)x0 x3 x9not (x0 x4 x9)not (x0 x5 x9)x0 x6 x9not (x0 x7 x9)x0 x8 x9x10)x10
Definition 94275.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 . ∀ x11 : ο . (7db3a.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9(x1 = x10∀ x12 : ο . x12)(x2 = x10∀ x12 : ο . x12)(x3 = x10∀ x12 : ο . x12)(x4 = x10∀ x12 : ο . x12)(x5 = x10∀ x12 : ο . x12)(x6 = x10∀ x12 : ο . x12)(x7 = x10∀ x12 : ο . x12)(x8 = x10∀ x12 : ο . x12)(x9 = x10∀ x12 : ο . x12)x0 x1 x10x0 x2 x10not (x0 x3 x10)not (x0 x4 x10)x0 x5 x10not (x0 x6 x10)not (x0 x7 x10)not (x0 x8 x10)x0 x9 x10x11)x11
Definition ce66c.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ο . (94275.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10(x1 = x11∀ x13 : ο . x13)(x2 = x11∀ x13 : ο . x13)(x3 = x11∀ x13 : ο . x13)(x4 = x11∀ x13 : ο . x13)(x5 = x11∀ x13 : ο . x13)(x6 = x11∀ x13 : ο . x13)(x7 = x11∀ x13 : ο . x13)(x8 = x11∀ x13 : ο . x13)(x9 = x11∀ x13 : ο . x13)(x10 = x11∀ x13 : ο . x13)not (x0 x1 x11)x0 x2 x11not (x0 x3 x11)not (x0 x4 x11)not (x0 x5 x11)not (x0 x6 x11)not (x0 x7 x11)x0 x8 x11not (x0 x9 x11)not (x0 x10 x11)x12)x12
Definition ac9f0.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . ∀ x13 : ο . (ce66c.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11(x1 = x12∀ x14 : ο . x14)(x2 = x12∀ x14 : ο . x14)(x3 = x12∀ x14 : ο . x14)(x4 = x12∀ x14 : ο . x14)(x5 = x12∀ x14 : ο . x14)(x6 = x12∀ x14 : ο . x14)(x7 = x12∀ x14 : ο . x14)(x8 = x12∀ x14 : ο . x14)(x9 = x12∀ x14 : ο . x14)(x10 = x12∀ x14 : ο . x14)(x11 = x12∀ x14 : ο . x14)x0 x1 x12not (x0 x2 x12)not (x0 x3 x12)not (x0 x4 x12)not (x0 x5 x12)x0 x6 x12not (x0 x7 x12)not (x0 x8 x12)not (x0 x9 x12)not (x0 x10 x12)x0 x11 x12x13)x13
Definition andand := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Definition nInnIn := λ x0 x1 . not (x0x1)
Known setminusEsetminusE : ∀ x0 x1 x2 . x2setminus x0 x1and (x2x0) (nIn x2 x1)
Known c7834.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3x1∀ x4 . x4x1x2 x3 x4x2 x4 x3)4402e.. x1 x2cf2df.. x1 x2∀ x3 . x3x1x0setminus x1 (Sing x3)∀ x4 . x4x0∀ x5 . x5x0∀ x6 . x6x0∀ x7 . x7x0∀ x8 . x8x0∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x0∀ x13 . x13x0∀ x14 . x14x00e688.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14∀ x15 : ο . (x2 x4 x3not (x2 x5 x3)not (x2 x6 x3)not (x2 x7 x3)x2 x8 x3not (x2 x9 x3)not (x2 x10 x3)not (x2 x11 x3)not (x2 x12 x3)not (x2 x13 x3)x2 x14 x3x15)(x2 x4 x3not (x2 x5 x3)not (x2 x6 x3)not (x2 x7 x3)x2 x8 x3x2 x9 x3not (x2 x10 x3)not (x2 x11 x3)not (x2 x12 x3)not (x2 x13 x3)x2 x14 x3x15)x15
Known neq_i_symneq_i_sym : ∀ x0 x1 . (x0 = x1∀ x2 : ο . x2)x1 = x0∀ x2 : ο . x2
Known Subq_traSubq_tra : ∀ x0 x1 x2 . x0x1x1x2x0x2
Known setminus_Subqsetminus_Subq : ∀ x0 x1 . setminus x0 x1x0
Known SingISingI : ∀ x0 . x0Sing x0
Theorem c6fe1.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3x1∀ x4 . x4x1x2 x3 x4x2 x4 x3)4402e.. x1 x2cf2df.. x1 x2∀ x3 . x3x1x0setminus x1 (Sing x3)∀ x4 . x4x0∀ x5 . x5x0∀ x6 . x6x0∀ x7 . x7x0∀ x8 . x8x0∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x0∀ x13 . x13x0∀ x14 . x14x00e688.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14∀ x15 : ο . (∀ x16 . x16x0∀ x17 . x17x0∀ x18 . x18x0∀ x19 . x19x0∀ x20 . x20x0∀ x21 . x21x0∀ x22 . x22x0∀ x23 . x23x0∀ x24 . x24x0∀ x25 . x25x0∀ x26 . x26x0f3db1.. x2 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x3 x26x15)(∀ x16 . x16x0∀ x17 . x17x0∀ x18 . x18x0∀ x19 . x19x0∀ x20 . x20x0∀ x21 . x21x0∀ x22 . x22x0∀ x23 . x23x0∀ x24 . x24x0∀ x25 . x25x0∀ x26 . x26x0ac9f0.. x2 x16 x17 x18 x19 x20 x21 x22 x23 x24 x3 x25 x26x15)x15 (proof)

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