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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.
Claim L4: ...
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Apply L4 with ∃ x3 . and (∃ x4 : ι → ο . PNo_least_rep2 x0 x1 x3 x4) (∀ x4 x5 : ι → ο . PNo_least_rep2 x0 x1 x3 x4 ⟶ PNo_least_rep2 x0 x1 x3 x5 ⟶ x4 = x5).
Let x3 of type ι be given.
Assume H5: (λ x4 . and (and (ordinal x4) (∃ x5 : ι → ο . PNo_strict_imv x0 x1 x4 x5)) (∀ x5 . x5 ∈ x4 ⟶ not (∃ x6 : ι → ο . PNo_strict_imv x0 x1 x5 x6))) x3.
Apply H5 with ∃ x4 . and (∃ x5 : ι → ο . PNo_least_rep2 x0 x1 x4 x5) (∀ x5 x6 : ι → ο . PNo_least_rep2 x0 x1 x4 x5 ⟶ PNo_least_rep2 x0 x1 x4 x6 ⟶ x5 = x6).
Assume H6: and (ordinal x3) (∃ x4 : ι → ο . PNo_strict_imv x0 x1 x3 x4).
Apply H6 with (∀ x4 . x4 ∈ x3 ⟶ not (∃ x5 : ι → ο . PNo_strict_imv x0 x1 x4 x5)) ⟶ ∃ x4 . and (∃ x5 : ι → ο . PNo_least_rep2 x0 x1 x4 x5) (∀ x5 x6 : ι → ο . PNo_least_rep2 x0 x1 x4 x5 ⟶ PNo_least_rep2 x0 x1 x4 x6 ⟶ x5 = x6).
Assume H7: ordinal x3.
Assume H8: ∃ x4 : ι → ο . PNo_strict_imv x0 x1 x3 x4.
Assume H9: ∀ x4 . x4 ∈ x3 ⟶ not (∃ x5 : ι → ο . PNo_strict_imv x0 x1 x4 x5).
Apply H8 with ∃ x4 . and (∃ x5 : ι → ο . PNo_least_rep2 x0 x1 x4 x5) (∀ x5 x6 : ι → ο . PNo_least_rep2 x0 x1 x4 x5 ⟶ PNo_least_rep2 x0 x1 x4 x6 ⟶ x5 = x6).
Let x4 of type ι → ο be given.
Assume H10: PNo_strict_imv x0 x1 x3 x4.
Let x5 of type ο be given.
Assume H11: ∀ x6 . and (∃ x7 : ι → ο . PNo_least_rep2 x0 x1 x6 x7) (∀ x7 x8 : ι → ο . PNo_least_rep2 x0 x1 x6 x7 ⟶ PNo_least_rep2 x0 x1 x6 x8 ⟶ x7 = x8) ⟶ x5.
Apply H11 with x3.
Apply andI with ∃ x6 : ι → ο . PNo_least_rep2 x0 x1 x3 x6, ∀ x6 x7 : ι → ο . PNo_least_rep2 x0 x1 x3 x6 ⟶ PNo_least_rep2 x0 x1 x3 x7 ⟶ x6 = x7 leaving 2 subgoals.
Let x6 of type ο be given.
Assume H12: ∀ x7 : ι → ο . PNo_least_rep2 x0 x1 x3 x7 ⟶ x6.
Apply H12 with λ x7 . and (x7 ∈ x3) (x4 x7).
Apply andI with PNo_least_rep x0 x1 x3 (λ x7 . and (x7 ∈ x3) (x4 x7)), ∀ x7 . nIn x7 x3 ⟶ not (and (x7 ∈ x3) (x4 x7)) leaving 2 subgoals.
Apply and3I with ordinal ..., ..., ... leaving 3 subgoals.
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