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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: 40dde.. x0 x3 x1 x4.
Apply unknownprop_1c12738cd89f8c615a541c15b6797bba2a5be97ab5e514c9fd76b3fef06e2aa9 with x0, x1, x3, x4, 40dde.. x0 x2 x1 x4 leaving 4 subgoals.
The subproof is completed by applying H3.
Assume H4: PNoLt_ (d3786.. x0 x1) x3 x4.
Apply H4 with 40dde.. x0 x2 x1 x4.
Let x5 of type ι be given.
Assume H5: (λ x6 . and (prim1 x6 (d3786.. x0 x1)) (and (and (PNoEq_ x6 x3 x4) (not (x3 x6))) (x4 x6))) x5.
Apply H5 with 40dde.. x0 x2 x1 x4.
Assume H6: prim1 x5 (d3786.. x0 x1).
Apply unknownprop_1ac99d32a7ae5dc08fd640ea6c8b661df6b3535fe85e88b30b17c3701cb4c7ce with x0, x1, x5, and (and (PNoEq_ x5 x3 x4) (not (x3 x5))) (x4 x5) ⟶ 40dde.. x0 x2 x1 x4 leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H7: prim1 x5 x0.
Assume H8: prim1 x5 x1.
Assume H9: and (and (PNoEq_ x5 x3 x4) (not (x3 x5))) (x4 x5).
Apply H9 with 40dde.. x0 x2 x1 x4.
Assume H10: and (PNoEq_ x5 x3 x4) (not (x3 x5)).
Apply H10 with x4 x5 ⟶ 40dde.. x0 x2 x1 x4.
Assume H11: PNoEq_ x5 x3 x4.
Assume H12: not (x3 x5).
Assume H13: x4 x5.
Apply unknownprop_ac970f51deca19d20e9a8350c3518ea802533ebd2768fe799c9b97ea3dd03596 with x0, x1, x2, x4.
Let x6 of type ο be given.
Assume H14: ∀ x7 . and (prim1 x7 (d3786.. x0 x1)) (and (and (PNoEq_ x7 x2 x4) (not (x2 x7))) (x4 x7)) ⟶ x6.
Apply H14 with x5.
Apply andI with prim1 x5 (d3786.. 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: prim1 x0 x1.
Assume H5: PNoEq_ x0 x3 x4.
Assume H6: x4 x0.
Apply unknownprop_9bcd3ca68066c7069c00444b0b53fe3ae6267cb29974000b72e5fe8327360c0b 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: prim1 x1 x0.
Assume H5: PNoEq_ x1 x3 x4.
Assume H6: not (x3 x1).
Apply unknownprop_b51c738b3a14385af55eef02a445728dc056a37996fdc42e5ede8e064af23c97 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 ..., ... leaving 2 subgoals.
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