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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0 ∈ prim4 3.
Let x1 of type ο be given.
Assume H1: x0 = 0 ⟶ x1.
Assume H2: x0 = 1 ⟶ x1.
Assume H3: x0 = Sing 1 ⟶ x1.
Assume H4: x0 = 2 ⟶ x1.
Assume H5: x0 = Sing 2 ⟶ x1.
Assume H6: x0 = UPair 0 2 ⟶ x1.
Assume H7: x0 = UPair 1 2 ⟶ x1.
Assume H8: x0 = 3 ⟶ x1.
Apply In_Power_ordsucc_cases_impred with 2, x0, x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H9: x0 ∈ prim4 2.
Apply In_Power_2_cases_impred with x0, x1 leaving 5 subgoals.
The subproof is completed by applying H9.
Assume H10: x0 = 0.
Apply H1.
The subproof is completed by applying H10.
Assume H10: x0 = 1.
Apply H2.
The subproof is completed by applying H10.
Assume H10: x0 = Sing 1.
Apply H3.
The subproof is completed by applying H10.
Assume H10: x0 = 2.
Apply H4.
The subproof is completed by applying H10.
Assume H9: 2 ∈ x0.
Assume H10: setminus x0 (Sing 2) ∈ prim4 2.
Apply In_Power_2_cases_impred with setminus x0 (Sing 2), x1 leaving 5 subgoals.
The subproof is completed by applying H10.
Assume H11: setminus x0 (Sing 2) = 0.
Apply H5.
Apply set_ext with x0, Sing 2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H12: x2 ∈ x0.
Apply dneg with x2 ∈ Sing 2.
Assume H13: nIn x2 (Sing 2).
Apply EmptyE with x2.
Apply H11 with λ x3 x4 . x2 ∈ x3.
Apply setminusI with x0, Sing 2, x2 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
Let x2 of type ι be given.
Assume H12: x2 ∈ Sing 2.
Apply SingE with 2, x2, λ x3 x4 . x4 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H9.
Assume H11: setminus x0 (Sing 2) = 1.
Apply H6.
Apply set_ext with x0, UPair 0 2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H12: x2 ∈ x0.
Apply cases_3 with x2, λ x3 . x3 ∈ x0 ⟶ x3 ∈ UPair 0 2 leaving 5 subgoals.
Apply PowerE with 3, x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H12.
Assume H13: 0 ∈ x0.
The subproof is completed by applying UPairI1 with 0, 2.
Assume H13: 1 ∈ x0.
Apply FalseE with 1 ∈ UPair 0 2.
Apply In_irref with 1.
Apply H11 with λ x3 x4 . 1 ∈ x3.
Apply setminusI with x0, Sing 2, 1 leaving 2 subgoals.
The subproof is completed by applying H13.
Assume H14: 1 ∈ Sing 2.
Apply neq_1_2.
Apply SingE with 2, 1.
The subproof is completed by applying H14.
Assume H13: 2 ∈ x0.
The subproof is completed by applying UPairI2 with 0, 2.
The subproof is completed by applying H12.
Let x2 of type ι be given.
Assume H12: x2 ∈ UPair 0 2.
Apply UPairE with x2, 0, 2, x2 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H12.
Assume H13: x2 = 0.
Apply setminusE1 with x0, Sing 2, x2.
Apply H11 with λ x3 x4 . x2 ∈ x4.
Apply H13 with λ x3 x4 . x4 ∈ 1.
The subproof is completed by applying In_0_1.
Assume H13: x2 = 2.
Apply H13 with λ x3 x4 . x4 ∈ x0.
The subproof is completed by applying H9.
Assume H11: setminus x0 (Sing 2) = Sing 1.
Apply H7.
Apply set_ext with x0, UPair 1 2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H12: x2 ∈ x0.
Apply cases_3 with x2, λ x3 . x3 ∈ x0 ⟶ x3 ∈ UPair 1 2 leaving 5 subgoals.
Apply PowerE with 3, x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H12.
Assume H13: 0 ∈ x0.
Apply FalseE with ....
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