Let x0 of type ι be given.
Assume H0: 0 ∈ x0.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H1: x1 ⊆ x2.
Apply Pi_0_dom_mon with
x1,
x2,
λ x3 . x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x2.
The subproof is completed by applying H0.