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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: ordinal x0.
Assume H1: ordinal x1.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Let x4 of type ι → ο be given.
Assume H2: PNoEq_ x0 x2 x3.
Assume H3: PNoLt x0 x3 x1 x4.
Apply PNoLtE with x0, x1, x3, x4, PNoLt x0 x2 x1 x4 leaving 4 subgoals.
The subproof is completed by applying H3.
Assume H4: PNoLt_ (binintersect x0 x1) x3 x4.
Apply H4 with PNoLt x0 x2 x1 x4.
Let x5 of type ι be given.
Assume H5: (λ x6 . and (x6 ∈ binintersect x0 x1) (and (and (PNoEq_ x6 x3 x4) (not (x3 x6))) (x4 x6))) x5.
Apply H5 with PNoLt x0 x2 x1 x4.
Assume H6: x5 ∈ binintersect x0 x1.
Apply binintersectE with x0, x1, x5, and (and (PNoEq_ x5 x3 x4) (not (x3 x5))) (x4 x5) ⟶ PNoLt x0 x2 x1 x4 leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H7: x5 ∈ x0.
Assume H8: x5 ∈ x1.
Assume H9: and (and (PNoEq_ x5 x3 x4) (not (x3 x5))) (x4 x5).
Apply H9 with PNoLt x0 x2 x1 x4.
Assume H10: and (PNoEq_ x5 x3 x4) (not (x3 x5)).
Apply H10 with x4 x5 ⟶ PNoLt x0 x2 x1 x4.
Assume H11: PNoEq_ x5 x3 x4.
Assume H12: not (x3 x5).
Assume H13: x4 x5.
Apply PNoLtI1 with x0, x1, x2, x4.
Let x6 of type ο be given.
Assume H14: ∀ x7 . and (x7 ∈ binintersect x0 x1) (and (and (PNoEq_ x7 x2 x4) (not (x2 x7))) (x4 x7)) ⟶ x6.
Apply H14 with x5.
Apply andI with x5 ∈ binintersect x0 x1, and (and (PNoEq_ x5 x2 x4) (not (x2 x5))) (x4 x5) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply and3I with PNoEq_ x5 x2 x4, not (x2 x5), x4 x5 leaving 3 subgoals.
Apply PNoEq_tra_ with x5, x2, x3, x4 leaving 2 subgoals.
Apply PNoEq_antimon_ with x2, x3, x0, x5 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H7.
The subproof is completed by applying H2.
The subproof is completed by applying H11.
Assume H15: x2 x5.
Apply H12.
Apply iffEL with x2 x5, x3 x5 leaving 2 subgoals.
Apply H2 with x5.
The subproof is completed by applying H7.
The subproof is completed by applying H15.
The subproof is completed by applying H13.
Assume H4: x0 ∈ x1.
Assume H5: PNoEq_ x0 x3 x4.
Assume H6: x4 x0.
Apply PNoLtI2 with x0, x1, x2, x4 leaving 3 subgoals.
The subproof is completed by applying H4.
Apply PNoEq_tra_ with x0, x2, x3, x4 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Assume H4: x1 ∈ x0.
Assume H5: PNoEq_ x1 x3 x4.
Assume H6: not (x3 x1).
Apply PNoLtI3 with x0, x1, x2, x4 leaving 3 subgoals.
The subproof is completed by applying H4.
Apply PNoEq_tra_ with x1, x2, x3, x4 leaving 2 subgoals.
Apply PNoEq_antimon_ with x2, x3, x0, x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Assume H7: x2 x1.
Apply H6.
Apply iffEL with x2 x1, x3 x1 leaving 2 subgoals.
Apply H2 with x1.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
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