Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H0:
∀ x8 . x8 ∈ x4 ⟶ ∀ x9 : ο . (x8 ∈ x0 ⟶ x8 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ (x3 = x8 ⟶ ∀ x10 : ο . x10) ⟶ x1 x3 x8 ⟶ (x8 = x2 ⟶ ∀ x10 : ο . x10) ⟶ x9) ⟶ x9.
Assume H1:
∃ x8 . and (x8 ∈ x4) (x1 x5 x8).
Assume H2:
x1 x5 x6 ⟶ x1 x7 x6 ⟶ ∀ x8 . x8 ∈ x4 ⟶ not (x1 x5 x8).
Assume H3: x1 x5 x6.
Assume H4: x1 x7 x6.
Apply H1 with
False.
Let x8 of type ι be given.
Assume H5:
(λ x9 . and (x9 ∈ x4) (x1 x5 x9)) x8.
Apply H5 with
False.
Assume H6: x8 ∈ x4.
Assume H7: x1 x5 x8.
Apply H0 with
x8,
False leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H8: x8 ∈ x0.
Assume H10: x3 = x8 ⟶ ∀ x9 : ο . x9.
Assume H11: x1 x3 x8.
Assume H12: x8 = x2 ⟶ ∀ x9 : ο . x9.
Apply H2 with
x8 leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
The subproof is completed by applying H7.