Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Apply H0 with
equip x0 x1.
Assume H2: ∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1.
Assume H3: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5.
Apply H1 with
equip x0 x1.
Assume H4: ∀ x4 . x4 ∈ x1 ⟶ x3 x4 ∈ x0.
Assume H5: ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5.
Apply KnasterTarski_set with
x0,
λ x4 . {x3 x5|x5 ∈ setminus x1 {x2 x5|x5 ∈ setminus x0 x4}},
equip x0 x1 leaving 3 subgoals.
The subproof is completed by applying L6.
The subproof is completed by applying L7.
Let x4 of type ι be given.
Apply H8 with
equip x0 x1.
Assume H9:
x4 ∈ prim4 x0.
Assume H10:
(λ x5 . {x3 x6|x6 ∈ setminus x1 {x2 x6|x6 ∈ setminus x0 x5}}) x4 = x4.
Let x5 of type ο be given.
Assume H11:
∀ x6 : ι → ι . bij x0 x1 x6 ⟶ x5.
Apply H11 with
λ x6 . If_i (x6 ∈ x4) (inv x1 x3 x6) (x2 x6).
Apply bijI with
x0,
x1,
λ x6 . If_i (x6 ∈ x4) (inv x1 x3 x6) (x2 x6) leaving 3 subgoals.
Let x6 of type ι be given.
Assume H12: x6 ∈ x0.
Apply xm with
x6 ∈ x4,
If_i (x6 ∈ x4) (inv x1 x3 x6) (x2 x6) ∈ x1 leaving 2 subgoals.
Assume H13: x6 ∈ x4.
Apply If_i_1 with
x6 ∈ x4,
inv x1 x3 x6,
x2 x6,
λ x7 x8 . x8 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H13.
Claim L14:
x6 ∈ (λ x7 . {x3 x8|x8 ∈ setminus x1 {x2 x8|x8 ∈ setminus x0 x7}}) x4Apply H10 with
λ x7 x8 . x6 ∈ x8.
The subproof is completed by applying H13.
Apply ReplE_impred with
setminus x1 {x2 x7|x7 ∈ setminus x0 x4},
x3,
x6,
inv x1 x3 x6 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying L14.
Let x7 of type ι be given.
Assume H16: x6 = x3 x7.
Claim L17: x7 ∈ x1
Apply setminusE1 with
x1,
{x2 x8|x8 ∈ setminus x0 x4},
x7.
The subproof is completed by applying H15.
Apply H16 with
λ x8 x9 . inv x1 x3 x9 ∈ x1.
Apply inj_linv_coddep with
x1,
x0,
x3,
x7,
λ x8 x9 . x9 ∈ x1 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying L17.
The subproof is completed by applying L17.