Let x0 of type ι → ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply set_ext with
{x0 x3|x3 ∈ UPair x1 x2},
UPair (x0 x1) (x0 x2) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0:
x3 ∈ {x0 x4|x4 ∈ UPair x1 x2}.
Apply ReplE_impred with
UPair x1 x2,
x0,
x3,
x3 ∈ UPair (x0 x1) (x0 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H1:
x4 ∈ UPair x1 x2.
Assume H2: x3 = x0 x4.
Apply H2 with
λ x5 x6 . x6 ∈ UPair (x0 x1) (x0 x2).
Apply UPairE with
x4,
x1,
x2,
x0 x4 ∈ UPair (x0 x1) (x0 x2) leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H3: x4 = x1.
Apply H3 with
λ x5 x6 . x0 x6 ∈ UPair (x0 x1) (x0 x2).
The subproof is completed by applying UPairI1 with x0 x1, x0 x2.
Assume H3: x4 = x2.
Apply H3 with
λ x5 x6 . x0 x6 ∈ UPair (x0 x1) (x0 x2).
The subproof is completed by applying UPairI2 with x0 x1, x0 x2.
Let x3 of type ι be given.
Assume H0:
x3 ∈ UPair (x0 x1) (x0 x2).
Apply UPairE with
x3,
x0 x1,
x0 x2,
x3 ∈ {x0 x4|x4 ∈ UPair x1 x2} leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 = x0 x1.
Apply H1 with
λ x4 x5 . x5 ∈ {x0 x6|x6 ∈ UPair x1 x2}.
Apply ReplI with
UPair x1 x2,
x0,
x1.
The subproof is completed by applying UPairI1 with x1, x2.
Assume H1: x3 = x0 x2.
Apply H1 with
λ x4 x5 . x5 ∈ {x0 x6|x6 ∈ UPair x1 x2}.
Apply ReplI with
UPair x1 x2,
x0,
x2.
The subproof is completed by applying UPairI2 with x1, x2.