Let x0 of type ι → (ι → ο) → ο be given.
Let x1 of type ι → (ι → ο) → ο be given.
Let x2 of type ι be given.
Apply H1 with
PNo_lenbdd x2 x0 ⟶ PNo_lenbdd x2 x1 ⟶ ∃ x3 . and (x3 ∈ ordsucc x2) (∃ x4 : ι → ο . PNo_rel_strict_split_imv x0 x1 x3 x4).
Assume H3:
∀ x3 . x3 ∈ x2 ⟶ TransSet x3.
Apply PNo_rel_imv_ex with
x0,
x1,
x2,
∃ x3 . and (x3 ∈ ordsucc x2) (∃ x4 : ι → ο . PNo_rel_strict_split_imv x0 x1 x3 x4) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply H7 with
∃ x3 . and (x3 ∈ ordsucc x2) (∃ x4 : ι → ο . PNo_rel_strict_split_imv x0 x1 x3 x4).
Let x3 of type ι → ο be given.
Apply H8 with
∃ x4 . and (x4 ∈ ordsucc x2) (∃ x5 : ι → ο . PNo_rel_strict_split_imv x0 x1 x4 x5).
Apply H9 with
(∀ x4 : ι → ο . PNo_rel_strict_imv x0 x1 x2 x4 ⟶ PNoEq_ x2 x3 x4) ⟶ ∃ x4 . and (x4 ∈ ordsucc x2) (∃ x5 : ι → ο . PNo_rel_strict_split_imv x0 x1 x4 x5).
Let x4 of type ο be given.
Apply H13 with
x2.
Apply andI with
x2 ∈ ordsucc x2,
∃ x5 : ι → ο . PNo_rel_strict_split_imv x0 x1 x2 x5 leaving 2 subgoals.
The subproof is completed by applying ordsuccI2 with x2.
Let x5 of type ο be given.
Apply H14 with
x3.
Apply PNo_lenbdd_strict_imv_split with
x0,
x1,
x2,
x3 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H9.