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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: nIn x2 x1.
Assume H1: equip x0 x1.
Apply H1 with equip (ordsucc x0) (binunion x1 (Sing x2)).
Let x3 of type ι → ι be given.
Assume H2: bij x0 x1 x3.
Apply bijE with x0, x1, x3, equip (ordsucc x0) (binunion x1 (Sing x2)) leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: ∀ x4 . x4 ∈ x0 ⟶ x3 x4 ∈ x1.
Assume H4: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5.
Assume H5: ∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x3 x5 = x4).
Let x4 of type ο be given.
Assume H6: ∀ x5 : ι → ι . bij (ordsucc x0) (binunion x1 (Sing x2)) x5 ⟶ x4.
Apply H6 with λ x5 . If_i (x5 ∈ x0) (x3 x5) x2.
Apply bijI with ordsucc x0, binunion x1 (Sing x2), λ x5 . If_i (x5 ∈ x0) (x3 x5) x2 leaving 3 subgoals.
Let x5 of type ι be given.
Assume H7: x5 ∈ ordsucc x0.
Apply ordsuccE with x0, x5, (λ x6 . If_i (x6 ∈ x0) (x3 x6) x2) x5 ∈ binunion x1 (Sing x2) leaving 3 subgoals.
The subproof is completed by applying H7.
Assume H8: x5 ∈ x0.
Apply If_i_1 with x5 ∈ x0, x3 x5, x2, λ x6 x7 . x7 ∈ binunion x1 (Sing x2) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply binunionI1 with x1, Sing x2, x3 x5.
Apply H3 with x5.
The subproof is completed by applying H8.
Assume H8: x5 = x0.
Claim L9: nIn x5 x0
Apply H8 with λ x6 x7 . nIn x7 x0.
The subproof is completed by applying In_irref with x0.
Apply If_i_0 with x5 ∈ x0, x3 x5, x2, λ x6 x7 . x7 ∈ binunion x1 (Sing x2) leaving 2 subgoals.
The subproof is completed by applying L9.
Apply binunionI2 with x1, Sing x2, x2.
The subproof is completed by applying SingI with x2.
Let x5 of type ι be given.
Assume H7: x5 ∈ ordsucc x0.
Let x6 of type ι be given.
Assume H8: x6 ∈ ordsucc x0.
Apply ordsuccE with x0, x5, If_i (x5 ∈ x0) (x3 x5) x2 = If_i (x6 ∈ x0) (x3 x6) x2 ⟶ x5 = x6 leaving 3 subgoals.
The subproof is completed by applying H7.
Assume H9: x5 ∈ x0.
Apply If_i_1 with x5 ∈ x0, x3 x5, x2, λ x7 x8 . x8 = If_i (x6 ∈ x0) (x3 x6) x2 ⟶ x5 = x6 leaving 2 subgoals.
The subproof is completed by applying H9.
Apply ordsuccE with x0, x6, x3 x5 = If_i (x6 ∈ x0) (x3 x6) x2 ⟶ x5 = x6 leaving 3 subgoals.
The subproof is completed by applying H8.
Assume H10: x6 ∈ x0.
Apply If_i_1 with x6 ∈ x0, x3 x6, x2, λ x7 x8 . x3 x5 = x8 ⟶ x5 = x6 leaving 2 subgoals.
The subproof is completed by applying H10.
Apply H4 with x5, x6 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Assume H10: x6 = x0.
Claim L11: ...
...
Apply If_i_0 with x6 ∈ x0, x3 x6, x2, λ x7 x8 . x3 ... = ... ⟶ x5 = x6 leaving 2 subgoals.
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