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.
Let x4 of type ι → ο be given.
Assume H1: x4 x0.
Assume H2: x4 x1.
Assume H3: x4 x2.
Apply binunionE with
UPair x0 x1,
Sing x2,
x3,
x4 x3 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H4:
x3 ∈ UPair x0 x1.
Apply UPairE with
x3,
x0,
x1,
x4 x3 leaving 3 subgoals.
The subproof is completed by applying H4.
Assume H5: x3 = x0.
Apply H5 with
λ x5 x6 . x4 x6.
The subproof is completed by applying H1.
Assume H5: x3 = x1.
Apply H5 with
λ x5 x6 . x4 x6.
The subproof is completed by applying H2.
Apply SingE with
x2,
x3,
λ x5 x6 . x4 x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.