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Proofgold Proof

pf
Let x0 of type ι(ιο) → ο be given.
Let x1 of type ι(ιο) → ο be given.
Let x2 of type ι be given.
Assume H0: ordinal x2.
Let x3 of type ιο be given.
Assume H1: ∀ x4 . prim1 x4 x2x3 x4.
Assume H2: ∀ x4 . ordinal x4∀ x5 : ι → ο . x0 x4 x5prim1 x4 x2.
Assume H3: ∀ x4 . prim1 x4 x2x0 x4 x3.
Assume H4: ∀ x4 . ordinal x4∀ x5 : ι → ο . not (x1 x4 x5).
Apply andI with cae4c.. x0 x2 x3, bc2b0.. x1 x2 x3 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H5: ordinal x4.
Let x5 of type ιο be given.
Assume H6: x0 x4 x5.
Claim L7: prim1 x4 x2
Apply H2 with x4, x5 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply unknownprop_73b6444bcb1b9cb998566f55e286e78644e785a99d955b3281cf269899ab486c with x2, x4, x3, x5, 40dde.. x4 x5 x2 x3 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Assume H8: or (40dde.. x2 x3 x4 x5) (and (x2 = x4) (PNoEq_ x2 x3 x5)).
Apply H8 with 40dde.. x4 x5 x2 x3 leaving 2 subgoals.
Assume H9: 40dde.. x2 x3 x4 x5.
Apply FalseE with 40dde.. x4 x5 x2 x3.
Apply unknownprop_1c12738cd89f8c615a541c15b6797bba2a5be97ab5e514c9fd76b3fef06e2aa9 with x2, x4, x3, x5, False leaving 4 subgoals.
The subproof is completed by applying H9.
Assume H10: PNoLt_ (d3786.. x2 x4) x3 x5.
Apply PNoLt_E_ with d3786.. x2 x4, x3, x5, False leaving 2 subgoals.
The subproof is completed by applying H10.
Let x6 of type ι be given.
Assume H11: prim1 x6 (d3786.. x2 x4).
Apply unknownprop_1ac99d32a7ae5dc08fd640ea6c8b661df6b3535fe85e88b30b17c3701cb4c7ce with x2, x4, x6, PNoEq_ x6 x3 x5not (x3 x6)x5 x6False leaving 2 subgoals.
The subproof is completed by applying H11.
Assume H12: prim1 x6 x2.
Assume H13: prim1 x6 x4.
Assume H14: PNoEq_ x6 x3 x5.
Assume H15: not (x3 x6).
Apply FalseE with x5 x6False.
Apply H15.
Apply H1 with x6.
The subproof is completed by applying H12.
Assume H10: prim1 x2 x4.
Apply FalseE with PNoEq_ x2 x3 x5x5 x2False.
Apply unknownprop_f1a526a64fd91875cd825eea7f2e7776b7f0e7be5dcee74dc03af1d7886d1eb6 with x2, x4 leaving 2 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying L7.
Assume H10: prim1 x4 x2.
Assume H11: PNoEq_ x4 x3 x5.
Assume H12: not (x3 x4).
Apply H12.
Apply H1 with x4.
The subproof is completed by applying L7.
Assume H9: and (x2 = x4) (PNoEq_ x2 x3 x5).
Apply H9 with 40dde.. x4 x5 x2 x3.
Assume H10: x2 = x4.
Apply FalseE with PNoEq_ x2 x3 x540dde.. x4 x5 x2 x3.
Apply In_irref with x2.
Apply H10 with λ x6 x7 . prim1 x7 x2.
The subproof is completed by applying L7.
Assume H8: 40dde.. x4 x5 x2 x3.
The subproof is completed by applying H8.
Let x4 of type ι be given.
Assume H5: ordinal x4.
Let x5 of type ιο be given.
Assume H6: x1 x4 x5.
Apply FalseE with 40dde.. x2 x3 x4 x5.
Apply H4 with x4, x5 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.