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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ⊆ prim4 (ordsucc x0).
Assume H1: ∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ ∃ x4 . and (x4 ∈ ordsucc x0) (x4 ∈ x2 = x4 ∈ x3).
Claim L2: ...
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Let x2 of type ο be given.
Assume H3: ∀ x3 : ι → ι . inj x1 (prim4 x0) x3 ⟶ x2.
Apply H3 with λ x3 . If_i (x0 ∈ x3) (setminus (ordsucc x0) x3) x3.
Apply andI with ∀ x3 . x3 ∈ x1 ⟶ (λ x4 . If_i (x0 ∈ x4) (setminus (ordsucc x0) x4) x4) x3 ∈ prim4 x0, ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ (λ x5 . If_i (x0 ∈ x5) (setminus (ordsucc x0) x5) x5) x3 = (λ x5 . If_i (x0 ∈ x5) (setminus (ordsucc x0) x5) x5) x4 ⟶ x3 = x4 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H4: x3 ∈ x1.
Apply xm with x0 ∈ x3, (λ x4 . If_i (x0 ∈ x4) (setminus (ordsucc x0) x4) x4) x3 ∈ prim4 x0 leaving 2 subgoals.
Assume H5: x0 ∈ x3.
Apply If_i_1 with x0 ∈ x3, setminus (ordsucc x0) x3, x3, λ x4 x5 . x5 ∈ prim4 x0 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply PowerI with x0, setminus (ordsucc x0) x3.
Let x4 of type ι be given.
Assume H6: x4 ∈ setminus (ordsucc x0) x3.
Apply setminusE with ordsucc x0, x3, x4, x4 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H7: x4 ∈ ordsucc x0.
Assume H8: nIn x4 x3.
Apply ordsuccE with x0, x4, x4 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H7.
Assume H9: x4 ∈ x0.
The subproof is completed by applying H9.
Assume H9: x4 = x0.
Apply FalseE with x4 ∈ x0.
Apply H8.
Apply H9 with λ x5 x6 . x6 ∈ x3.
The subproof is completed by applying H5.
Assume H5: nIn x0 x3.
Apply If_i_0 with x0 ∈ x3, setminus (ordsucc x0) x3, x3, λ x4 x5 . x5 ∈ prim4 x0 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply PowerI with x0, x3.
Let x4 of type ι be given.
Assume H6: x4 ∈ x3.
Apply ordsuccE with x0, x4, x4 ∈ x0 leaving 3 subgoals.
Apply L2 with x3, x4 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
Assume H7: x4 ∈ x0.
The subproof is completed by applying H7.
Assume H7: x4 = x0.
Apply FalseE with x4 ∈ x0.
Apply H5.
Apply H7 with λ x5 x6 . x5 ∈ x3.
The subproof is completed by applying H6.
Let x3 of type ι be given.
Assume H4: x3 ∈ x1.
Let x4 of type ι be given.
Assume H5: x4 ∈ x1.
Apply xm with x0 ∈ x3, (λ x5 . If_i (x0 ∈ x5) (setminus (ordsucc x0) x5) x5) x3 = (λ x5 . If_i (x0 ∈ x5) (setminus (ordsucc x0) x5) x5) x4 ⟶ x3 = x4 leaving 2 subgoals.
Assume H6: x0 ∈ x3.
Apply If_i_1 with ..., ..., ..., ... leaving 2 subgoals.
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