Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Assume H1:
∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).
Assume H2:
∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).
Let x1 of type ι be given.
Apply equip_sym with
u6,
binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1).
Apply unknownprop_eab190d6552dbda6c7d00c3e93c1ad9385698a8d73462a2a4e5795b67701610d with
u5,
DirGraphOutNeighbors u18 x0 x1,
x1 leaving 2 subgoals.
Apply SepE2 with
u18,
λ x2 . and (x1 = x2 ⟶ ∀ x3 : ο . x3) (x0 x1 x2),
x1,
False leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H5: x1 = x1 ⟶ ∀ x2 : ο . x2.
Apply FalseE with
x0 x1 x1 ⟶ False.
Apply H5.
Let x2 of type ι → ι → ο be given.
Assume H6: x2 x1 x1.
The subproof is completed by applying H6.
Apply equip_sym with
DirGraphOutNeighbors u18 x0 x1,
u5.
Apply unknownprop_942eb02a74c10f16602e1ae1f344c3023e05004e91bcaa34f489f7c49867be93 with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.