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