Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H0:
{x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ∈ prim4 x0.
Assume H1:
x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ∈ prim4 x0.
Assume H2:
∀ x2 . x2 ∈ prim4 x0 ⟶ x1 x2 ⊆ x2 ⟶ {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} ⊆ x2.
Assume H3:
x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ⊆ {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3}.
Assume H4:
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}.
Let x2 of type ο be given.
Assume H5:
∀ x3 . and (x3 ∈ prim4 x0) (x1 x3 = x3) ⟶ x2.
Apply H5 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 H0.
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 H3.
Apply H2 with
x1 {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.