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.
Let x2 of type ι be given.
Apply unknownprop_997b324045b1165b0cf38788927ff324ddb3a505c8b91e290586e4dd5480f2bd with
DirGraphOutNeighbors u18 x0 x2,
u4,
x1 leaving 2 subgoals.
Apply unknownprop_426b271b8453605fe796f284fb15405fbff198d07b0c3dc7b8c218dee82a5c65 with
u18,
x0,
x1,
x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply unknownprop_942eb02a74c10f16602e1ae1f344c3023e05004e91bcaa34f489f7c49867be93 with
x0,
x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SepE1 with
u18,
λ x3 . and (x1 = x3 ⟶ ∀ x4 : ο . x4) (x0 x1 x3),
x2.
The subproof is completed by applying H4.