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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Assume H0: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3.
Assume H1: 4402e.. x1 x2.
Assume H2: cf2df.. x1 x2.
Let x3 of type ι be given.
Assume H3: x3 ∈ x1.
Assume H4: x0 ⊆ setminus x1 (Sing x3).
Let x4 of type ι be given.
Assume H5: x4 ∈ x0.
Let x5 of type ι be given.
Assume H6: x5 ∈ x0.
Let x6 of type ι be given.
Assume H7: x6 ∈ x0.
Let x7 of type ι be given.
Assume H8: x7 ∈ x0.
Let x8 of type ι be given.
Assume H9: x8 ∈ x0.
Let x9 of type ι be given.
Assume H10: x9 ∈ x0.
Let x10 of type ι be given.
Assume H11: x10 ∈ x0.
Let x11 of type ι be given.
Assume H12: x11 ∈ x0.
Apply setminusE with x1, Sing x3, x4, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x4.
The subproof is completed by applying H5.
Assume H13: x4 ∈ x1.
Assume H14: nIn x4 (Sing x3).
Apply setminusE with x1, Sing x3, x5, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x5.
The subproof is completed by applying H6.
Assume H15: x5 ∈ x1.
Assume H16: nIn x5 (Sing x3).
Apply setminusE with x1, Sing x3, x6, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x6.
The subproof is completed by applying H7.
Assume H17: x6 ∈ x1.
Assume H18: nIn x6 (Sing x3).
Apply setminusE with x1, Sing x3, x7, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x7.
The subproof is completed by applying H8.
Assume H19: x7 ∈ x1.
Assume H20: nIn x7 (Sing x3).
Apply setminusE with x1, Sing x3, x8, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x8.
The subproof is completed by applying H9.
Assume H21: x8 ∈ x1.
Assume H22: nIn x8 (Sing x3).
Apply setminusE with x1, Sing x3, x9, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x9.
The subproof is completed by applying H10.
Assume H23: x9 ∈ x1.
Assume H24: nIn x9 (Sing x3).
Apply setminusE with x1, Sing x3, x10, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x10.
The subproof is completed by applying H11.
Assume H25: x10 ∈ x1.
Assume H26: nIn x10 (Sing x3).
Apply setminusE with x1, Sing x3, x11, 372ed.. x2 x4 x5 x6 x7 x8 x9 x10 x11 ⟶ ∀ x12 : ο . x12 leaving 2 subgoals.
Apply H4 with x11.
The subproof is completed by applying H12.
Assume H27: x11 ∈ x1.
Assume H28: nIn x11 (Sing x3).
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Assume H37: 372ed.. x2 ... ... ... ... ... ... ... ....
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