Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Assume H2:
∀ x4 . x4 ⊆ x0 ⟶ infinite x4 ⟶ and (x2 x4 ∈ x4) (nIn (x2 x4) (x1 x4)).
Assume H3: x3 0 = x1 x0.
Claim L7:
∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ x4 ⟶ x2 (x3 x5) = x2 (x3 x4) ⟶ ∀ x6 : ο . x6Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H8: x5 ∈ x4.
Assume H9: x2 (x3 x5) = x2 (x3 x4).
Apply L5 with
x4,
False leaving 2 subgoals.
Apply omega_nat_p with
x4.
The subproof is completed by applying H7.
Assume H11: x3 x4 ⊆ x0.
Apply H2 with
x3 x4,
False leaving 3 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Assume H13: x2 (x3 x4) ∈ x3 x4.
Assume H14:
nIn (x2 (x3 x4)) (x1 (x3 x4)).
Apply L5 with
x5,
False leaving 2 subgoals.
The subproof is completed by applying L10.
Assume H15: x3 x5 ⊆ x0.
Apply H2 with
x3 x5,
False leaving 3 subgoals.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
Assume H17: x2 (x3 x5) ∈ x3 x5.
Assume H18:
nIn (x2 (x3 x5)) (x1 (x3 x5)).
Apply H18.
Apply H9 with
λ x6 x7 . x7 ∈ x1 (x3 x5).
Apply H4 with
x5,
λ x6 x7 . x3 x4 ⊆ x6,
x2 (x3 x4) leaving 3 subgoals.
The subproof is completed by applying L10.
Apply L6 with
ordsucc x5,
x4 leaving 3 subgoals.
Apply omega_ordsucc with
x5.
Apply nat_p_omega with
x5.
The subproof is completed by applying L10.
The subproof is completed by applying H7.
Apply ordinal_ordsucc_In_Subq with
x4,
x5 leaving 2 subgoals.
Apply nat_p_ordinal with
x4.
Apply omega_nat_p with
x4.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H13.
Apply and3I with
∀ x4 . nat_p x4 ⟶ and (x3 x4 ⊆ x0) (infinite (x3 x4)),
∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ omega ⟶ x4 ⊆ x5 ⟶ x3 x5 ⊆ x3 x4,
∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ omega ⟶ x2 (x3 x4) = x2 (x3 x5) ⟶ x4 = x5 leaving 3 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H10: x2 (x3 x4) = x2 (x3 x5).
Apply ordinal_trichotomy_or_impred with
x4,
x5,
x4 = x5 leaving 5 subgoals.
Apply nat_p_ordinal with
x4.
Apply omega_nat_p with
x4.
The subproof is completed by applying H8.
Apply nat_p_ordinal with
x5.
Apply omega_nat_p with
x5.
The subproof is completed by applying H9.
Assume H11: x4 ∈ x5.
Apply FalseE with
x4 = x5.
Apply L7 with
x5,
x4 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H11.
The subproof is completed by applying H10.
Assume H11: x4 = x5.
The subproof is completed by applying H11.
Assume H11: x5 ∈ x4.
Apply FalseE with
x4 = x5.
Apply L7 with
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H11.
Let x6 of type ι → ι → ο be given.
The subproof is completed by applying H10 with λ x7 x8 . x6 x8 x7.