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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: 1beb9.. x0 x1.
Apply H1 with x1 = 09072.. x0 (λ x2 . prim1 x2 x1).
Assume H2: Subq x1 (472ec.. x0).
Assume H3: ∀ x2 . prim1 x2 x0 ⟶ exactly1of2 (prim1 ((λ x3 . 15418.. x3 (91630.. (4ae4a.. 4a7ef..))) x2) x1) (prim1 x2 x1).
Apply set_ext with x1, 09072.. x0 (λ x2 . prim1 x2 x1) leaving 2 subgoals.
Let x2 of type ι be given.
Assume H4: prim1 x2 x1.
Apply unknownprop_b46721c187c37140cbae22d356b00ba89f4126d81d8665e4be15b5a58c78d06f with x0, 94f9e.. x0 (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3), x2, prim1 x2 (09072.. x0 (λ x3 . prim1 x3 x1)) leaving 3 subgoals.
Apply H2 with x2.
The subproof is completed by applying H4.
Assume H5: prim1 x2 x0.
Apply unknownprop_0b5b61a66a1ed2eb843dbce5c5f6930c63a284fe5e33704d9f0cc564310ec40b with 1216a.. x0 (λ x3 . prim1 x3 x1), a4c2a.. x0 (λ x3 . nIn x3 x1) (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3), x2.
Apply unknownprop_1dada0fb38ff7f9b45b564ad11d6295d93250427446875218f17ee62431454a6 with x0, λ x3 . prim1 x3 x1, x2 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
Assume H5: prim1 x2 (94f9e.. x0 (λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3)).
Apply unknownprop_d908b89102f7b662c739e5a844f67efc8ae1cd05a2e9ce1e3546fa3885f40100 with x0, λ x3 . (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3, x2, prim1 x2 (09072.. x0 (λ x3 . prim1 x3 x1)) leaving 2 subgoals.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Assume H6: prim1 x3 x0.
Assume H7: x2 = (λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3.
Apply unknownprop_e4d6e0bfb4ef6d52ee13edd54a77c8cc7f0a3af8ffb1b8da66d4f98842dd28b5 with 1216a.. x0 (λ x4 . prim1 x4 x1), a4c2a.. x0 (λ x4 . nIn x4 x1) (λ x4 . (λ x5 . 15418.. x5 (91630.. (4ae4a.. 4a7ef..))) x4), x2.
Apply H7 with λ x4 x5 . prim1 x5 (a4c2a.. x0 (λ x6 . nIn x6 x1) (λ x6 . (λ x7 . 15418.. x7 (91630.. (4ae4a.. 4a7ef..))) x6)).
Claim L8: nIn x3 x1
Assume H8: prim1 x3 x1.
Apply exactly1of2_E with prim1 ((λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3) x1, prim1 x3 x1, False leaving 3 subgoals.
Apply H3 with x3.
The subproof is completed by applying H6.
Assume H9: prim1 ((λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3) x1.
Assume H10: nIn x3 x1.
Apply H10.
The subproof is completed by applying H8.
Assume H9: nIn ((λ x4 . 15418.. x4 (91630.. (4ae4a.. 4a7ef..))) x3) x1.
Assume H10: prim1 x3 x1.
Apply H9.
Apply H7 with λ x4 x5 . prim1 x4 x1.
The subproof is completed by applying H4.
Apply unknownprop_9c424e90871cd9e3a05e6a3be208792c52ad17517e2db8ef40187e46ac0e9a6e with x0, λ x4 . nIn x4 x1, λ x4 . (λ x5 . 15418.. x5 (91630.. (4ae4a.. 4a7ef..))) x4, x3 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying L8.
Let x2 of type ι be given.
Assume H4: prim1 ... ....
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