Claim L0: ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ x0) ⟶ (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ (λ x3 . λ x4 : ι → ι . ∀ x5 . x5 ∈ x3 ⟶ x4 (x4 x5) = x4 x5) x0 x2 ⟶ (λ x3 . λ x4 : ι → ι . ∀ x5 . x5 ∈ x3 ⟶ x4 (x4 x5) = x4 x5) x0 x1
Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ x0.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3.
Assume H2: (λ x3 . λ x4 : ι → ι . ∀ x5 . x5 ∈ x3 ⟶ x4 (x4 x5) = x4 x5) x0 x2.
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Apply H1 with
x1 x3,
λ x4 x5 . x5 = x1 x3 leaving 2 subgoals.
Apply H0 with
x3.
The subproof is completed by applying H3.
Apply H1 with
x3,
λ x4 x5 . x2 x5 = x5 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H2 with
x3.
The subproof is completed by applying H3.
Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H1: ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0.
Let x2 of type ι → ι be given.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3.
Apply prop_ext_2 with
∀ x3 . x3 ∈ x0 ⟶ x2 (x2 x3) = x2 x3,
∀ x3 . x3 ∈ x0 ⟶ x1 (x1 x3) = x1 x3 leaving 2 subgoals.
Apply L0 with
x0,
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Assume H3: (λ x3 . λ x4 : ι → ι . ∀ x5 . x5 ∈ x3 ⟶ x4 (x4 x5) = x4 x5) x0 x1.
Apply L0 with
x0,
x2,
x1 leaving 3 subgoals.
Let x3 of type ι be given.
Assume H4: x3 ∈ x0.
Apply H2 with
x3,
λ x4 x5 . x4 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H1 with
x3.
The subproof is completed by applying H4.
Let x3 of type ι be given.
Assume H4: x3 ∈ x0.
Let x4 of type ι → ι → ο be given.
Apply H2 with
x3,
λ x5 x6 . x4 x6 x5.
The subproof is completed by applying H4.
The subproof is completed by applying H3.