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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ιο be given.
Let x3 of type ιο be given.
Assume H0: ordinal x0.
Assume H1: ordinal x1.
Apply H0 with or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2).
Assume H2: TransSet x0.
Assume H3: ∀ x4 . x4x0TransSet x4.
Apply H1 with or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2).
Assume H4: TransSet x1.
Assume H5: ∀ x4 . x4x1TransSet x4.
Claim L6: ...
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Apply PNoLt_trichotomy_or_ with x2, x3, binintersect x0 x1, or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 3 subgoals.
The subproof is completed by applying L6.
Assume H7: or (PNoLt_ (binintersect x0 x1) x2 x3) (PNoEq_ (binintersect x0 x1) x2 x3).
Apply H7 with or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Assume H8: PNoLt_ (binintersect x0 x1) x2 x3.
Apply or3I1 with PNoLt x0 x2 x1 x3, and (x0 = x1) (PNoEq_ x0 x2 x3), PNoLt x1 x3 x0 x2.
Apply PNoLtI1 with x0, x1, x2, x3.
The subproof is completed by applying H8.
Assume H8: PNoEq_ (binintersect x0 x1) x2 x3.
Apply ordinal_trichotomy_or with x0, x1, or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H9: or (x0x1) (x0 = x1).
Apply H9 with or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Assume H10: x0x1.
Claim L11: ...
...
Claim L12: ...
...
Apply xm with x3 x0, or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) leaving 2 subgoals.
Assume H13: x3 x0.
Apply or3I1 with PNoLt x0 x2 x1 x3, and (x0 = x1) (PNoEq_ x0 x2 x3), PNoLt x1 x3 x0 x2.
Apply PNoLtI2 with x0, x1, x2, x3 leaving 3 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying L12.
The subproof is completed by applying H13.
Assume H13: not (x3 ...).
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