Let x0 of type ι → ο be given.
Let x1 of type ι → ο be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x3 ∈ x2.
Apply H2 with
PNoLt_ x2 x0 x1.
Let x4 of type ι be given.
Assume H3:
(λ x5 . and (x5 ∈ x3) (and (and (PNoEq_ x5 x0 x1) (not (x0 x5))) (x1 x5))) x4.
Apply H3 with
PNoLt_ x2 x0 x1.
Assume H4: x4 ∈ x3.
Let x5 of type ο be given.
Assume H6:
∀ x6 . and (x6 ∈ x2) (and (and (PNoEq_ x6 x0 x1) (not (x0 x6))) (x1 x6)) ⟶ x5.
Apply H6 with
x4.
Apply andI with
x4 ∈ x2,
and (and (PNoEq_ x4 x0 x1) (not (x0 x4))) (x1 x4) leaving 2 subgoals.
Apply H0 with
x4 ∈ x2.
Assume H8:
∀ x6 . x6 ∈ x2 ⟶ TransSet x6.
Apply H7 with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H5.