Let x0 of type ι be given.
Apply H0 with
x0 ∈ setminus omega (Sing 0).
Apply H1 with
(∀ x1 . x1 ∈ omega ⟶ divides_nat x1 x0 ⟶ or (x1 = 1) (x1 = x0)) ⟶ x0 ∈ setminus omega (Sing 0).
Assume H3: 1 ∈ x0.
Apply setminusI with
omega,
Sing 0,
x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply In_no2cycle with
1,
x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply SingE with
0,
x0,
λ x1 x2 . x2 ∈ 1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying In_0_1.