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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0: x2x0.
Assume H1: equip x0 (ordsucc x1).
Apply H1 with equip (setminus x0 (Sing x2)) x1.
Let x3 of type ιι be given.
Assume H2: bij x0 (ordsucc x1) x3.
Apply bijE with x0, ordsucc x1, x3, equip (setminus x0 (Sing x2)) x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: ∀ x4 . x4x0x3 x4ordsucc x1.
Assume H4: ∀ x4 . x4x0∀ x5 . x5x0x3 x4 = x3 x5x4 = x5.
Assume H5: ∀ x4 . x4ordsucc x1∃ x5 . and (x5x0) (x3 x5 = x4).
Apply equip_tra with setminus x0 (Sing x2), setminus (ordsucc x1) (Sing (x3 x2)), x1 leaving 2 subgoals.
Let x4 of type ο be given.
Assume H6: ∀ x5 : ι → ι . bij (setminus x0 (Sing x2)) (setminus (ordsucc x1) (Sing (x3 x2))) x5x4.
Apply H6 with x3.
Apply unknownprop_20ec276501d9ecb91e40cc4525c5a2c0798ab59924056be0d591bc4dcbb72338 with x0, ordsucc x1, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply equip_sym with x1, setminus (ordsucc x1) (Sing (x3 x2)).
Let x4 of type ο be given.
Assume H6: ∀ x5 : ι → ι . bij x1 (setminus (ordsucc x1) (Sing (x3 x2))) x5x4.
Apply H6 with λ x5 . If_i (x5 = x3 x2) x1 x5.
Apply bijI with x1, setminus (ordsucc x1) (Sing (x3 x2)), λ x5 . If_i (x5 = x3 x2) x1 x5 leaving 3 subgoals.
Let x5 of type ι be given.
Assume H7: x5x1.
Apply xm with x5 = x3 x2, If_i (x5 = x3 x2) x1 x5setminus (ordsucc x1) (Sing (x3 x2)) leaving 2 subgoals.
Assume H8: x5 = x3 x2.
Apply If_i_1 with x5 = x3 x2, x1, x5, λ x6 x7 . x7setminus (ordsucc x1) (Sing (x3 x2)) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply setminusI with ordsucc x1, Sing (x3 x2), x1 leaving 2 subgoals.
The subproof is completed by applying ordsuccI2 with x1.
Assume H9: x1Sing (x3 x2).
Apply In_irref with x1.
Apply SingE with x3 x2, x1, λ x6 x7 . x7x1 leaving 2 subgoals.
The subproof is completed by applying H9.
Apply H8 with λ x6 x7 . x6x1.
The subproof is completed by applying H7.
Assume H8: not (x5 = x3 x2).
Apply If_i_0 with x5 = x3 x2, x1, x5, λ x6 x7 . x7setminus (ordsucc x1) (Sing (x3 x2)) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply setminusI with ordsucc x1, Sing (x3 x2), x5 leaving 2 subgoals.
Apply ordsuccI1 with x1, x5.
The subproof is completed by applying H7.
Assume H9: x5Sing (x3 x2).
Apply H8.
Apply SingE with x3 x2, x5.
The subproof is completed by applying H9.
Let x5 of type ι be given.
Assume H7: x5x1.
Let x6 of type ι be given.
Assume H8: x6x1.
Apply xm with x5 = x3 x2, If_i (x5 = x3 x2) ... ... = ...x5 = x6 leaving 2 subgoals.
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