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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ordinal x0.
Let x1 of type ιο be given.
Let x2 of type ιο be given.
Let x3 of type ιο be given.
Assume H1: PNoLt_ x0 x1 x2.
Assume H2: PNoLt_ x0 x2 x3.
Apply PNoLt_E_ with x0, x1, x2, ∃ x4 . and (x4x0) (and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H3: x4x0.
Assume H4: PNoEq_ x4 x1 x2.
Assume H5: not (x1 x4).
Assume H6: x2 x4.
Apply PNoLt_E_ with x0, x2, x3, ∃ x5 . and (x5x0) (and (and (PNoEq_ x5 x1 x3) (not (x1 x5))) (x3 x5)) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x5 of type ι be given.
Assume H7: x5x0.
Assume H8: PNoEq_ x5 x2 x3.
Assume H9: not (x2 x5).
Assume H10: x3 x5.
Claim L11: ...
...
Claim L12: ...
...
Apply ordinal_trichotomy_or with x4, x5, ∃ x6 . and (x6x0) (and (and (PNoEq_ x6 x1 x3) (not (x1 x6))) (x3 x6)) leaving 4 subgoals.
The subproof is completed by applying L11.
The subproof is completed by applying L12.
Assume H13: or (x4x5) (x4 = x5).
Apply H13 with ∃ x6 . and (x6x0) (and (and (PNoEq_ x6 x1 x3) (not (x1 x6))) (x3 x6)) leaving 2 subgoals.
Assume H14: x4x5.
Let x6 of type ο be given.
Assume H15: ∀ x7 . and (x7x0) (and (and (PNoEq_ x7 x1 x3) (not (x1 x7))) (x3 x7))x6.
Apply H15 with x4.
Apply andI with x4x0, and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply and3I with PNoEq_ x4 x1 x3, not (x1 x4), x3 x4 leaving 3 subgoals.
Apply PNoEq_tra_ with x4, x1, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply PNoEq_antimon_ with x2, x3, x5, x4 leaving 3 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying H14.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
Apply H8 with x4, x3 x4 leaving 2 subgoals.
The subproof is completed by applying H14.
Assume H16: x2 x4x3 x4.
Assume H17: x3 x4x2 x4.
Apply H16.
The subproof is completed by applying H6.
Assume H14: x4 = x5.
Let x6 of type ο be given.
Assume H15: ∀ x7 . and (x7x0) (and (and (PNoEq_ x7 x1 x3) (not (x1 x7))) (x3 x7))x6.
Apply H15 with x4.
Apply andI with x4x0, and (and (PNoEq_ x4 x1 x3) (not (x1 x4))) (x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply and3I with PNoEq_ x4 x1 x3, not (x1 x4), x3 x4 leaving 3 subgoals.
Apply PNoEq_tra_ with x4, x1, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H14 with λ x7 x8 . PNoEq_ x8 x2 x3.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
Apply H14 with λ x7 x8 . x3 x8.
The subproof is completed by applying H10.
Assume H13: x5x4.
Let x6 of type ο be given.
Assume H14: ∀ x7 . and (x7x0) (and (and ... ...) ...)x6.
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