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.
Apply binunionE with
UPair x0 x1,
Sing x2,
x3,
x3 ∈ binunion (UPair x0 x2) (Sing x1) leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1:
x3 ∈ UPair x0 x1.
Apply UPairE with
x3,
x0,
x1,
x3 ∈ binunion (UPair x0 x2) (Sing x1) leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H2: x3 = x0.
Apply H2 with
λ x4 x5 . x5 ∈ binunion (UPair x0 x2) (Sing x1).
Apply binunionI1 with
UPair x0 x2,
Sing x1,
x0.
The subproof is completed by applying UPairI1 with x0, x2.
Assume H2: x3 = x1.
Apply H2 with
λ x4 x5 . x5 ∈ binunion (UPair x0 x2) (Sing x1).
Apply binunionI2 with
UPair x0 x2,
Sing x1,
x1.
The subproof is completed by applying SingI with x1.
Apply SingE with
x2,
x3,
λ x4 x5 . x5 ∈ binunion (UPair x0 x2) (Sing x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply binunionI1 with
UPair x0 x2,
Sing x1,
x2.
The subproof is completed by applying UPairI2 with x0, x2.