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Proofgold Proof

pf
Let x0 of type ι(ιο) → ο be given.
Let x1 of type ι be given.
Assume H0: ordinal x1.
Let x2 of type ιο be given.
Let x3 of type ι be given.
Assume H1: x3x1.
Apply H0 with PNo_rel_strict_upperbd x0 x1 x2PNo_rel_strict_upperbd x0 x3 x2.
Assume H2: TransSet x1.
Assume H3: ∀ x4 . x4x1TransSet x4.
Claim L4: ordinal x3
Apply ordinal_Hered with x1, x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Claim L5: TransSet x3
Apply L4 with TransSet x3.
Assume H5: TransSet x3.
Assume H6: ∀ x4 . x4x3TransSet x4.
The subproof is completed by applying H5.
Assume H6: ∀ x4 . x4x1∀ x5 : ι → ο . PNo_downc x0 x4 x5PNoLt x4 x5 x1 x2.
Let x4 of type ι be given.
Assume H7: x4x3.
Let x5 of type ιο be given.
Assume H8: PNo_downc x0 x4 x5.
Claim L9: x4x1
Apply H2 with x3, x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
Claim L10: PNoLt x4 x5 x1 x2
Apply H6 with x4, x5 leaving 2 subgoals.
The subproof is completed by applying L9.
The subproof is completed by applying H8.
Apply PNoLtE with x4, x1, x5, x2, PNoLt x4 x5 x3 x2 leaving 4 subgoals.
The subproof is completed by applying L10.
Assume H11: PNoLt_ (binintersect x4 x1) x5 x2.
Claim L12: binintersect x4 x1 = x4
Apply binintersect_Subq_eq_1 with x4, x1.
Apply H2 with x4.
The subproof is completed by applying L9.
Claim L13: binintersect x4 x3 = x4
Apply binintersect_Subq_eq_1 with x4, x3.
Apply L5 with x4.
The subproof is completed by applying H7.
Apply PNoLtI1 with x4, x3, x5, x2.
Apply L13 with λ x6 x7 . PNoLt_ x7 x5 x2.
Apply L12 with λ x6 x7 . PNoLt_ x6 x5 x2.
The subproof is completed by applying H11.
Assume H11: x4x1.
Assume H12: PNoEq_ x4 x5 x2.
Assume H13: x2 x4.
Apply PNoLtI2 with x4, x3, x5, x2 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
Assume H11: x1x4.
Apply FalseE with PNoEq_ x1 x5 x2not (x5 x1)PNoLt x4 x5 x3 x2.
Apply In_no2cycle with x1, x4 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying L9.