Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ο be given.
Assume H0:
PNoLt x0 x1 x0 x2.
Apply PNoLtE with
x0,
x0,
x1,
x2,
PNoLt_ x0 x1 x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying binintersect_idem with
x0,
λ x3 x4 . PNoLt_ x3 x1 x2.
Assume H1: x0 ∈ x0.
Apply FalseE with
PNoEq_ x0 x1 x2 ⟶ x2 x0 ⟶ PNoLt_ x0 x1 x2.
Apply In_irref with
x0.
The subproof is completed by applying H1.
Assume H1: x0 ∈ x0.
Apply FalseE with
PNoEq_ x0 x1 x2 ⟶ not (x1 x0) ⟶ PNoLt_ x0 x1 x2.
Apply In_irref with
x0.
The subproof is completed by applying H1.