Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Apply H0 with
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2).
Assume H3:
∀ x4 . x4 ∈ x0 ⟶ TransSet x4.
Apply H1 with
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2).
Assume H5:
∀ x4 . x4 ∈ x1 ⟶ TransSet x4.
Apply PNoLt_trichotomy_or_ with
x2,
x3,
binintersect x0 x1,
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 3 subgoals.
The subproof is completed by applying L6.
Apply H7 with
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Apply or3I1 with
PNoLt x0 x2 x1 x3,
and (x0 = x1) (PNoEq_ x0 x2 x3),
PNoLt x1 x3 x0 x2.
Apply PNoLtI1 with
x0,
x1,
x2,
x3.
The subproof is completed by applying H8.
Apply ordinal_trichotomy_or with
x0,
x1,
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H9:
or (x0 ∈ x1) (x0 = x1).
Apply H9 with
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Assume H10: x0 ∈ x1.
Apply xm with
x3 x0,
or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Assume H13: x3 x0.
Apply or3I1 with
PNoLt x0 x2 x1 x3,
and (x0 = x1) (PNoEq_ x0 x2 x3),
PNoLt x1 x3 x0 x2.
Apply PNoLtI2 with
x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying L12.
The subproof is completed by applying H13.
Assume H13:
not (x3 ...).