Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Apply H0 with
ccad8.. x0 x1.
Assume H1:
and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4).
Apply H1 with
(∀ x3 . x3 ∈ x1 ⟶ ∃ x4 . and (x4 ∈ x0) (x2 x4 = x3)) ⟶ ccad8.. x0 x1.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1.
Assume H3: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Assume H4:
∀ x3 . x3 ∈ x1 ⟶ ∃ x4 . and (x4 ∈ x0) (x2 x4 = x3).
Let x3 of type ο be given.
Apply H5 with
{KPair_alt7 x4 (x2 x4)|x4 ∈ x0}.
Apply and3I with
∀ x4 . x4 ∈ x0 ⟶ ∃ x5 . and (x5 ∈ x1) (KPair_alt7 x4 x5 ∈ {KPair_alt7 x6 (x2 x6)|x6 ∈ x0}),
∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (KPair_alt7 x5 x4 ∈ {KPair_alt7 x6 (x2 x6)|x6 ∈ x0}),
∀ x4 x5 x6 x7 . KPair_alt7 x4 x5 ∈ {KPair_alt7 x8 (x2 x8)|x8 ∈ x0} ⟶ KPair_alt7 x6 x7 ∈ {KPair_alt7 x8 (x2 x8)|x8 ∈ x0} ⟶ iff (x4 = x6) (x5 = x7) leaving 3 subgoals.
Let x4 of type ι be given.
Assume H6: x4 ∈ x0.
Let x5 of type ο be given.
Apply H7 with
x2 x4.
Apply andI with
x2 x4 ∈ x1,
KPair_alt7 x4 (x2 x4) ∈ {KPair_alt7 x6 (x2 x6)|x6 ∈ x0} leaving 2 subgoals.
Apply H2 with
x4.
The subproof is completed by applying H6.
Apply ReplI with
x0,
λ x6 . KPair_alt7 x6 (x2 x6),
x4.
The subproof is completed by applying H6.
Let x4 of type ι be given.
Assume H6: x4 ∈ x1.