Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply UnionE_impred with
UPair x0 x1,
x2,
or (x2 ∈ x0) (x2 ∈ x1) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: x2 ∈ x3.
Assume H2:
x3 ∈ UPair x0 x1.
Apply UPairE with
x3,
x0,
x1,
or (x2 ∈ x0) (x2 ∈ x1) leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x3 = x0.
Apply orIL with
x2 ∈ x0,
x2 ∈ x1.
Apply H3 with
λ x4 x5 . x2 ∈ x4.
The subproof is completed by applying H1.
Assume H3: x3 = x1.
Apply orIR with
x2 ∈ x0,
x2 ∈ x1.
Apply H3 with
λ x4 x5 . x2 ∈ x4.
The subproof is completed by applying H1.