Let x0 of type ι → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι be given.
Assume H1: x3 ∈ x1.
Apply H0 with
PNo_rel_strict_lowerbd x0 x1 x2 ⟶ PNo_rel_strict_lowerbd x0 x3 x2.
Assume H3:
∀ x4 . x4 ∈ x1 ⟶ TransSet x4.
Apply ordinal_Hered with
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply L4 with
TransSet x3.
Assume H6:
∀ x4 . x4 ∈ x3 ⟶ TransSet x4.
The subproof is completed by applying H5.
Assume H6:
∀ x4 . x4 ∈ x1 ⟶ ∀ x5 : ι → ο . PNo_upc x0 x4 x5 ⟶ PNoLt x1 x2 x4 x5.
Let x4 of type ι be given.
Assume H7: x4 ∈ x3.
Let x5 of type ι → ο be given.
Claim L9: x4 ∈ x1
Apply H2 with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
Claim L10:
PNoLt x1 x2 x4 x5Apply H6 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying L9.
The subproof is completed by applying H8.
Apply PNoLtE with
x1,
x4,
x2,
x5,
PNoLt x3 x2 x4 x5 leaving 4 subgoals.
The subproof is completed by applying L10.
Apply binintersect_com with
x1,
x4,
λ x6 x7 . x7 = x4.
Apply binintersect_Subq_eq_1 with
x4,
x1.
Apply H2 with
x4.
The subproof is completed by applying L9.
Apply binintersect_com with
x3,
x4,
λ x6 x7 . x7 = x4.
Apply binintersect_Subq_eq_1 with
x4,
x3.
Apply L5 with
x4.
The subproof is completed by applying H7.
Apply PNoLtI1 with
x3,
x4,
x2,
x5.
Apply L13 with
λ x6 x7 . PNoLt_ x7 x2 x5.
Apply L12 with
λ x6 x7 . PNoLt_ x6 x2 x5.
The subproof is completed by applying H11.
Assume H11: x1 ∈ x4.
Apply FalseE with
PNoEq_ x1 x2 x5 ⟶ x5 x1 ⟶ PNoLt x3 x2 x4 x5.
Apply In_no2cycle with
x1,
x4 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying L9.
Assume H11: x4 ∈ x1.
Apply PNoLtI3 with
x3,
x4,
x2,
x5 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H12.
The subproof is completed by applying H13.