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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: equip x0 x1.
Apply H0 with equip (ordsucc x0) (Inj1 x1).
Let x2 of type ι → ι be given.
Assume H1: bij x0 x1 x2.
Apply bijE with x0, x1, x2, equip (ordsucc x0) (Inj1 x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1.
Assume H3: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Assume H4: ∀ x3 . x3 ∈ x1 ⟶ ∃ x4 . and (x4 ∈ x0) (x2 x4 = x3).
Let x3 of type ο be given.
Assume H5: ∀ x4 : ι → ι . bij (ordsucc x0) (Inj1 x1) x4 ⟶ x3.
Apply H5 with λ x4 . If_i (x4 ∈ x0) (Inj1 (x2 x4)) 0.
Apply bijI with ordsucc x0, Inj1 x1, λ x4 . If_i (x4 ∈ x0) (Inj1 (x2 x4)) 0 leaving 3 subgoals.
Let x4 of type ι be given.
Assume H6: x4 ∈ ordsucc x0.
Apply ordsuccE with x0, x4, (λ x5 . If_i (x5 ∈ x0) (Inj1 (x2 x5)) 0) x4 ∈ Inj1 x1 leaving 3 subgoals.
The subproof is completed by applying H6.
Assume H7: x4 ∈ x0.
Apply If_i_1 with x4 ∈ x0, Inj1 (x2 x4), 0, λ x5 x6 . x6 ∈ Inj1 x1 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply Inj1I2 with x1, x2 x4.
Apply H2 with x4.
The subproof is completed by applying H7.
Assume H7: x4 = x0.
Apply H7 with λ x5 x6 . If_i (x6 ∈ x0) (Inj1 (x2 x6)) 0 ∈ Inj1 x1.
Apply If_i_0 with x0 ∈ x0, Inj1 (x2 x0), 0, λ x5 x6 . x6 ∈ Inj1 x1 leaving 2 subgoals.
The subproof is completed by applying In_irref with x0.
The subproof is completed by applying Inj1I1 with x1.
Let x4 of type ι be given.
Assume H6: x4 ∈ ordsucc x0.
Let x5 of type ι be given.
Assume H7: x5 ∈ ordsucc x0.
Apply ordsuccE with x0, x4, (λ x6 . If_i (x6 ∈ x0) (Inj1 (x2 x6)) 0) x4 = (λ x6 . If_i (x6 ∈ x0) (Inj1 (x2 x6)) 0) x5 ⟶ x4 = x5 leaving 3 subgoals.
The subproof is completed by applying H6.
Assume H8: x4 ∈ x0.
Apply If_i_1 with x4 ∈ x0, Inj1 (x2 x4), 0, λ x6 x7 . x7 = (λ x8 . If_i (x8 ∈ x0) (Inj1 (x2 x8)) 0) x5 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H8.
Apply ordsuccE with x0, x5, Inj1 (x2 x4) = (λ x6 . If_i (x6 ∈ x0) (Inj1 (x2 x6)) 0) x5 ⟶ x4 = x5 leaving 3 subgoals.
The subproof is completed by applying H7.
Assume H9: x5 ∈ x0.
Apply If_i_1 with x5 ∈ x0, Inj1 (x2 x5), 0, λ x6 x7 . Inj1 (x2 x4) = x7 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H9.
Assume H10: Inj1 (x2 x4) = Inj1 (x2 x5).
Apply H3 with x4, x5 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply Inj1_inj with x2 x4, x2 x5.
The subproof is completed by applying H10.
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