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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: c2e41.. x0 x1.
Apply H0 with ∃ x2 : ι → ι . bij x0 x1 x2.
Let x2 of type ι be given.
Assume H1: (λ x3 . and (and (∀ x4 . prim1 x4 x0 ⟶ ∃ x5 . and (prim1 x5 x1) (prim1 (7ee77.. x4 x5) x3)) (∀ x4 . prim1 x4 x1 ⟶ ∃ x5 . and (prim1 x5 x0) (prim1 (7ee77.. x5 x4) x3))) (∀ x4 x5 x6 x7 . prim1 (7ee77.. x4 x5) x3 ⟶ prim1 (7ee77.. x6 x7) x3 ⟶ iff (x4 = x6) (x5 = x7))) x2.
Apply H1 with ∃ x3 : ι → ι . bij x0 x1 x3.
Assume H2: and (∀ x3 . prim1 x3 x0 ⟶ ∃ x4 . and (prim1 x4 x1) (prim1 (7ee77.. x3 x4) x2)) (∀ x3 . prim1 x3 x1 ⟶ ∃ x4 . and (prim1 x4 x0) (prim1 (7ee77.. x4 x3) x2)).
Apply H2 with (∀ x3 x4 x5 x6 . prim1 (7ee77.. x3 x4) x2 ⟶ prim1 (7ee77.. x5 x6) x2 ⟶ iff (x3 = x5) (x4 = x6)) ⟶ ∃ x3 : ι → ι . bij x0 x1 x3.
Assume H3: ∀ x3 . prim1 x3 x0 ⟶ ∃ x4 . and (prim1 x4 x1) (prim1 (7ee77.. x3 x4) x2).
Assume H4: ∀ x3 . prim1 x3 x1 ⟶ ∃ x4 . and (prim1 x4 x0) (prim1 (7ee77.. x4 x3) x2).
Assume H5: ∀ x3 x4 x5 x6 . prim1 (7ee77.. x3 x4) x2 ⟶ prim1 (7ee77.. x5 x6) x2 ⟶ iff (x3 = x5) (x4 = x6).
Claim L6: ...
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Claim L7: ...
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Claim L8: ...
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Let x3 of type ο be given.
Assume H9: ∀ x4 : ι → ι . and (and (∀ x5 . prim1 x5 x0 ⟶ prim1 (x4 x5) x1) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 = x4 x6 ⟶ x5 = x6)) (∀ x5 . prim1 x5 x1 ⟶ ∃ x6 . and (prim1 x6 x0) (x4 x6 = x5)) ⟶ x3.
Apply H9 with λ x4 . prim0 (λ x5 . prim1 (7ee77.. x4 x5) x2).
Apply and3I with ∀ x4 . prim1 x4 x0 ⟶ prim1 ((λ x5 . prim0 (λ x6 . prim1 (7ee77.. x5 x6) x2)) x4) x1, ∀ x4 . ... ⟶ ∀ x5 . prim1 x5 ... ⟶ (λ x6 . prim0 (λ x7 . prim1 (7ee77.. x6 x7) x2)) x4 = (λ x6 . prim0 (λ x7 . prim1 (7ee77.. x6 x7) x2)) x5 ⟶ x4 = x5, ... leaving 3 subgoals.
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