Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Assume H0: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3.
Let x3 of type ι be given.
Assume H2: x3 ∈ x1.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Let x5 of type ι be given.
Assume H5: x5 ∈ x0.
Let x6 of type ι be given.
Assume H6: x6 ∈ x0.
Let x7 of type ι be given.
Assume H7: x7 ∈ x0.
Apply setminusE with
x1,
Sing x3,
x4,
a1d37.. x2 x4 x5 x6 x7 ⟶ ∀ x8 : ο . (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 32385.. x2 x9 x10 x11 x12 x3 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 0ee5a.. x2 x9 x3 x10 x11 x12 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 22dda.. x2 x9 x10 x3 x11 x12 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 87c36.. x2 x3 x9 x10 x11 x12 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 42d13.. x2 x9 x3 x10 x11 x12 ⟶ x8) ⟶ x8 leaving 2 subgoals.
Apply H3 with
x4.
The subproof is completed by applying H4.
Assume H8: x4 ∈ x1.
Apply setminusE with
x1,
Sing x3,
x5,
... ⟶ ∀ x8 : ο . ... ⟶ (∀ x9 . ... ⟶ ∀ x10 . ... ⟶ ∀ x11 . ... ⟶ ∀ x12 . ... ⟶ 0ee5a.. x2 x9 x3 x10 ... ... ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 22dda.. x2 x9 x10 x3 x11 x12 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 87c36.. x2 x3 x9 x10 x11 x12 ⟶ x8) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ 42d13.. x2 x9 x3 x10 x11 x12 ⟶ x8) ⟶ x8 leaving 2 subgoals.