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
ccad8.. 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
ccad8.. 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}},
ccad8.. 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
ccad8.. x0 x1.
Assume H9:
x4 ∈ prim4 x0.
Assume H10:
(λ x5 . {x3 x6|x6 ∈ setminus x1 {x2 x6|x6 ∈ setminus x0 x5}}) x4 = x4.
Apply unknownprop_20d9cb9439a9eb14b8db57b56040cdfb7bc1872855c5076ea7f74fca016ec44d with
x0,
x1,
λ x5 . If_i (x5 ∈ x4) (inv x1 x3 x5) (x2 x5).
Apply and3I with
∀ x5 . x5 ∈ x0 ⟶ (λ x6 . If_i (x6 ∈ x4) (inv x1 x3 x6) (x2 x6)) x5 ∈ x1,
∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ (λ x7 . If_i (x7 ∈ x4) (inv x1 x3 x7) (x2 x7)) x5 = (λ x7 . If_i (x7 ∈ x4) (inv x1 x3 x7) (x2 x7)) x6 ⟶ x5 = x6,
∀ x5 . x5 ∈ x1 ⟶ ∃ x6 . and (x6 ∈ x0) ((λ x7 . If_i (x7 ∈ x4) (inv x1 x3 x7) (x2 x7)) x6 = x5) leaving 3 subgoals.
Let x5 of type ι be given.
Assume H11: x5 ∈ x0.
Apply xm with
x5 ∈ x4,
If_i (x5 ∈ x4) (inv x1 x3 x5) (x2 x5) ∈ x1 leaving 2 subgoals.
Assume H12: x5 ∈ x4.
Apply If_i_1 with
x5 ∈ x4,
inv x1 x3 x5,
x2 x5,
λ x6 x7 . x7 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H12.
Apply ReplE_impred with
setminus x1 {x2 x6|x6 ∈ setminus x0 x4},
x3,
x5,
inv x1 x3 x5 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying L13.
Let x6 of type ι be given.