Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Assume H1:
∀ x1 . x1 ⊆ u17 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).
Assume H2:
∀ x1 . x1 ⊆ u17 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).
Apply unknownprop_affc49913747fa11b095e9305cedbaaa950055db35b087439dc7fd718eda5a78 with
x0 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 TwoRamseyProp_atleastp_atleastp with
u3,
u3,
u6,
u6,
u17 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying atleastp_ref with
u3.
The subproof is completed by applying atleastp_ref with
u6.