Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H1: ∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2.
Assume H2:
∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4).
Assume H3:
∀ x2 . x2 ⊆ x0 ⟶ atleastp u5 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4)).
Let x2 of type ι be given.
Assume H4: x2 ∈ x0.
Apply unknownprop_45d11dce2d0b092bd17c01d64c29c5885c90b43dc7cb762c6d6ada999ea508c5 with
u3,
DirGraphOutNeighbors x0 x1 x2,
atleastp u4 (DirGraphOutNeighbors x0 x1 x2) leaving 3 subgoals.
The subproof is completed by applying nat_3.
Apply FalseE with
atleastp u4 (DirGraphOutNeighbors x0 x1 x2).
Apply L8 with
x1,
False leaving 3 subgoals.
The subproof is completed by applying H1.
Apply H9 with
False.
Let x3 of type ι be given.
Apply H10 with
False.
Assume H12:
and (equip u3 x3) (∀ x4 . x4 ∈ x3 ⟶ ∀ x5 . x5 ∈ x3 ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x4 x5).
Apply H12 with
False.
Assume H14: ∀ x4 . x4 ∈ x3 ⟶ ∀ x5 . x5 ∈ x3 ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x4 x5.
Apply equip_atleastp with
u3,
x3.
The subproof is completed by applying H13.
Apply H2 with
x3 leaving 3 subgoals.
The subproof is completed by applying L15.
The subproof is completed by applying L16.
The subproof is completed by applying H14.