Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H0:
∀ x2 . x2 ∈ prim4 x0 ⟶ x1 x2 ∈ prim4 x0.
Assume H1:
∀ x2 . x2 ∈ prim4 x0 ⟶ ∀ x3 . x3 ∈ prim4 x0 ⟶ x2 ⊆ x3 ⟶ x1 x2 ⊆ x1 x3.
Claim L5:
x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ⊆ {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3}Let x2 of type ι be given.
Assume H5:
x2 ∈ x1 {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4}.
Apply SepI with
x0,
λ x3 . ∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4,
x2 leaving 2 subgoals.
Apply PowerE with
x0,
x1 {x3 ∈ x0|∀ x4 . x4 ∈ ... ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4},
... leaving 2 subgoals.
Claim L6:
x1 (x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3}) ⊆ x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3}Apply H1 with
x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3},
{x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L2.
The subproof is completed by applying L5.
Let x2 of type ο be given.
Assume H7:
∀ x3 . and (x3 ∈ prim4 x0) (x1 x3 = x3) ⟶ x2.
Apply H7 with
{x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4}.
Apply andI with
{x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} ∈ prim4 x0,
x1 {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} = {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} leaving 2 subgoals.
The subproof is completed by applying L2.
Apply set_ext with
x1 {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4},
{x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} leaving 2 subgoals.
The subproof is completed by applying L5.
Apply L4 with
x1 {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L6.