Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Apply H0 with
False.
Assume H1:
∀ x2 . x2 ∈ prim4 x0 ⟶ x1 x2 ∈ x0.
Assume H2:
∀ x2 . x2 ∈ prim4 x0 ⟶ ∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Apply PowerI with
x0,
{x1 x2|x2 ∈ prim4 x0,nIn (x1 x2) x2}.
Let x2 of type ι be given.
Assume H3:
x2 ∈ {x1 x3|x3 ∈ prim4 x0,nIn (x1 x3) x3}.
Apply ReplSepE_impred with
prim4 x0,
λ x3 . nIn (x1 x3) x3,
x1,
x2,
x2 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4:
x3 ∈ prim4 x0.
Assume H5:
nIn (x1 x3) x3.
Assume H6: x2 = x1 x3.
Apply H6 with
λ x4 x5 . x5 ∈ x0.
Apply H1 with
x3.
The subproof is completed by applying H4.
Apply ReplSepE_impred with
prim4 x0,
λ x2 . nIn (x1 x2) x2,
x1,
x1 {x1 x2|x2 ∈ prim4 x0,nIn (x1 x2) x2},
False leaving 2 subgoals.
The subproof is completed by applying H4.
Let x2 of type ι be given.
Assume H5:
x2 ∈ prim4 x0.
Assume H6:
nIn (x1 x2) x2.
Assume H7:
x1 {x1 x3|x3 ∈ prim4 x0,nIn (x1 x3) x3} = x1 x2.
Claim L8:
{x1 x3|x3 ∈ prim4 x0,nIn (x1 x3) x3} = x2Apply H2 with
{x1 x3|x3 ∈ prim4 x0,nIn (x1 x3) x3},
x2 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply H6.
Apply L8 with
λ x3 x4 . x1 x3 ∈ x3.
The subproof is completed by applying H4.
Apply L4.
Apply ReplSepI with
prim4 x0,
λ x2 . nIn (x1 x2) x2,
x1,
{x1 x2|x2 ∈ prim4 x0,nIn (x1 x2) x2} leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L4.