Let x0 of type ι be given.
Assume H0:
∀ x1 . x1 ∈ x0 ⟶ SNo x1.
Apply and3I with
∀ x1 . x1 ∈ x0 ⟶ SNo x1,
∀ x1 . x1 ∈ 0 ⟶ SNo x1,
∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ 0 ⟶ SNoLt x1 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Assume H1: x1 ∈ 0.
Apply FalseE with
SNo x1.
Apply EmptyE with
x1.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H1: x1 ∈ x0.
Let x2 of type ι be given.
Assume H2: x2 ∈ 0.
Apply FalseE with
SNoLt x1 x2.
Apply EmptyE with
x2.
The subproof is completed by applying H2.