Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
The subproof is completed by applying UPairI2 with
UPair x2 x3,
Sing x2.
Apply H0 with
λ x4 x5 . Sing x2 ∈ x5.
The subproof is completed by applying L1.
Apply UPairE with
Sing x2,
UPair x0 x1,
Sing x0,
x0 = x2 leaving 3 subgoals.
The subproof is completed by applying L2.
Apply Sing_inv with
x2,
UPair x0 x1,
x0 = x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4:
x2 ∈ UPair x0 x1.
Assume H5:
∀ x4 . x4 ∈ UPair x0 x1 ⟶ x4 = x2.
Apply H5 with
x0.
The subproof is completed by applying UPairI1 with x0, x1.
Apply Sing_inv with
x2,
Sing x0,
x0 = x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H5:
∀ x4 . x4 ∈ Sing x0 ⟶ x4 = x2.
Apply H5 with
x0.
The subproof is completed by applying SingI with x0.