Let x0 of type ι be given.
Apply and3I with
∀ x1 . x1 ∈ Sing x0 ⟶ SNo x1,
∀ x1 . x1 ∈ SNoR x0 ⟶ SNo x1,
∀ x1 . x1 ∈ Sing x0 ⟶ ∀ x2 . x2 ∈ SNoR x0 ⟶ SNoLt x1 x2 leaving 3 subgoals.
Let x1 of type ι be given.
Apply SingE with
x0,
x1,
λ x2 x3 . SNo x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Apply SNoR_E with
x0,
x1,
SNo x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNoR_E with
x0,
x2,
SNoLt x1 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SingE with
x0,
x1,
λ x3 x4 . SNoLt x4 x2.
The subproof is completed by applying H1.