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Proofgold Proof

pf
Let x0 of type ι → (ι → ο) → ο be given.
Let x1 of type ι → (ι → ο) → ο be given.
Assume H0: PNoLt_pwise x0 x1.
Let x2 of type ι be given.
Assume H1: ordinal x2.
Assume H2: PNo_lenbdd x2 x0.
Assume H3: PNo_lenbdd x2 x1.
Apply PNo_bd_pred with x0, x1, x2, PNo_bd x0 x1 ∈ ordsucc x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H4: and (ordinal (PNo_bd x0 x1)) (PNo_strict_imv x0 x1 (PNo_bd x0 x1) (PNo_pred x0 x1)).
Apply H4 with (∀ x3 . x3 ∈ PNo_bd x0 x1 ⟶ ∀ x4 : ι → ο . not (PNo_strict_imv x0 x1 x3 x4)) ⟶ PNo_bd x0 x1 ∈ ordsucc x2.
Assume H5: ordinal (PNo_bd x0 x1).
Assume H6: PNo_strict_imv x0 x1 (PNo_bd x0 x1) (PNo_pred x0 x1).
Assume H7: ∀ x3 . x3 ∈ PNo_bd x0 x1 ⟶ ∀ x4 : ι → ο . not (PNo_strict_imv x0 x1 x3 x4).
Apply PNo_lenbdd_strict_imv_ex with x0, x1, x2, PNo_bd x0 x1 ∈ ordsucc x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H8: (λ x4 . and (x4 ∈ ordsucc x2) (∃ x5 : ι → ο . PNo_strict_imv x0 x1 x4 x5)) x3.
Apply H8 with PNo_bd x0 x1 ∈ ordsucc x2.
Assume H9: x3 ∈ ordsucc x2.
Assume H10: ∃ x4 : ι → ο . PNo_strict_imv x0 x1 x3 x4.
Apply H10 with PNo_bd x0 x1 ∈ ordsucc x2.
Let x4 of type ι → ο be given.
Assume H11: PNo_strict_imv x0 x1 x3 x4.
Claim L12: ordinal (ordsucc x2)
Apply ordinal_ordsucc with x2.
The subproof is completed by applying H1.
Apply ordinal_In_Or_Subq with PNo_bd x0 x1, ordsucc x2, PNo_bd x0 x1 ∈ ordsucc x2 leaving 4 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying L12.
Assume H13: PNo_bd x0 x1 ∈ ordsucc x2.
The subproof is completed by applying H13.
Assume H13: ordsucc x2 ⊆ PNo_bd x0 x1.
Apply FalseE with PNo_bd x0 x1 ∈ ordsucc x2.
Claim L14: x3 ∈ PNo_bd x0 x1
Apply H13 with x3.
The subproof is completed by applying H9.
Apply H7 with x3, x4 leaving 2 subgoals.
The subproof is completed by applying L14.
The subproof is completed by applying H11.
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