Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Apply PNoLt_E_ with
x0,
x1,
x2,
∃ x4 . and (x4 ∈ x0) (and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Assume H6: x2 x4.
Apply PNoLt_E_ with
x0,
x2,
x3,
∃ x5 . and (x5 ∈ x0) (and (and (PNoEq_ x5 x1 x3) (not (x1 x5))) (x3 x5)) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x5 of type ι be given.
Assume H7: x5 ∈ x0.
Assume H10: x3 x5.
Apply ordinal_trichotomy_or with
x4,
x5,
∃ x6 . and (x6 ∈ x0) (and (and (PNoEq_ x6 x1 x3) (not (x1 x6))) (x3 x6)) leaving 4 subgoals.
The subproof is completed by applying L11.
The subproof is completed by applying L12.
Assume H13:
or (x4 ∈ x5) (x4 = x5).
Apply H13 with
∃ x6 . and (x6 ∈ x0) (and (and (PNoEq_ x6 x1 x3) (not (x1 x6))) (x3 x6)) leaving 2 subgoals.
Assume H14: x4 ∈ x5.
Let x6 of type ο be given.
Assume H15:
∀ x7 . and (x7 ∈ x0) (and (and (PNoEq_ x7 x1 x3) (not (x1 x7))) (x3 x7)) ⟶ x6.
Apply H15 with
x4.
Apply andI with
x4 ∈ x0,
and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply and3I with
PNoEq_ x4 x1 x3,
not (x1 x4),
x3 x4 leaving 3 subgoals.
Apply PNoEq_tra_ with
x4,
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply PNoEq_antimon_ with
x2,
x3,
x5,
x4 leaving 3 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying H14.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
Apply H8 with
x4,
x3 x4 leaving 2 subgoals.
The subproof is completed by applying H14.
Assume H16: x2 x4 ⟶ x3 x4.
Assume H17: x3 x4 ⟶ x2 x4.
Apply H16.
The subproof is completed by applying H6.
Assume H14: x4 = x5.
Let x6 of type ο be given.
Assume H15:
∀ x7 . and (x7 ∈ x0) (and (and (PNoEq_ x7 x1 x3) (not (x1 x7))) (x3 x7)) ⟶ x6.
Apply H15 with
x4.
Apply andI with
x4 ∈ x0,
and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply and3I with
PNoEq_ x4 x1 x3,
not (x1 x4),
x3 x4 leaving 3 subgoals.
Apply PNoEq_tra_ with
x4,
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H14 with
λ x7 x8 . PNoEq_ x8 x2 x3.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
Apply H14 with
λ x7 x8 . x3 x8.
The subproof is completed by applying H10.
Assume H13: x5 ∈ x4.
Let x6 of type ο be given.
Assume H14:
∀ x7 . and (x7 ∈ x0) (and (and ... ...) ...) ⟶ x6.