Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Apply H0 with
atleastp (binintersect x1 x0) (binintersect {x2 x3|x3 ∈ x1} x0).
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Let x3 of type ο be given.
Apply H3 with
x2.
Apply andI with
∀ x4 . x4 ∈ binintersect x1 x0 ⟶ x2 x4 ∈ binintersect {x2 x5|x5 ∈ x1} x0,
∀ x4 . x4 ∈ binintersect x1 x0 ⟶ ∀ x5 . x5 ∈ binintersect x1 x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5 leaving 2 subgoals.
Let x4 of type ι be given.
Apply binintersectE with
x1,
x0,
x4,
x2 x4 ∈ binintersect {x2 x5|x5 ∈ x1} x0 leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H5: x4 ∈ x1.
Assume H6: x4 ∈ x0.
Apply binintersectI with
{x2 x5|x5 ∈ x1},
x0,
x2 x4 leaving 2 subgoals.
Apply ReplI with
x1,
x2,
x4.
The subproof is completed by applying H5.
Apply H1 with
x4.
The subproof is completed by applying H6.
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply binintersectE with
x1,
x0,
x4,
x2 x4 = x2 x5 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H6: x4 ∈ x1.
Assume H7: x4 ∈ x0.
Apply binintersectE with
x1,
x0,
x5,
x2 x4 = x2 x5 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H8: x5 ∈ x1.
Assume H9: x5 ∈ x0.
Apply H2 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H9.