Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Assume H0: x0 ∈ x1.
Assume H2: x3 x0.
Apply or3I2 with
PNoLt_ (binintersect x0 x1) x2 x3,
and (and (x0 ∈ x1) (PNoEq_ x0 x2 x3)) (x3 x0),
and (and (x1 ∈ x0) (PNoEq_ x1 x2 x3)) (not (x2 x1)).
Apply and3I with
x0 ∈ x1,
PNoEq_ x0 x2 x3,
x3 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.