Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0:
x0 ∈ UPair x1 x2.
Apply ReplE with
prim4 (prim4 0),
λ x3 . If_i (0 ∈ x3) x1 x2,
x0,
or (x0 = x1) (x0 = x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Claim L2:
x0 = If_i (0 ∈ x3) x1 x2Apply andER with
x3 ∈ prim4 (prim4 0),
x0 = If_i (0 ∈ x3) x1 x2.
The subproof is completed by applying H1.
Apply If_i_or with
0 ∈ x3,
x1,
x2,
or (x0 = x1) (x0 = x2) leaving 2 subgoals.
Assume H3:
If_i (0 ∈ x3) x1 x2 = x1.
Apply orIL with
x0 = x1,
x0 = x2.
Apply H3 with
λ x4 x5 . x0 = x4.
The subproof is completed by applying L2.
Assume H3:
If_i (0 ∈ x3) x1 x2 = x2.
Apply orIR with
x0 = x1,
x0 = x2.
Apply H3 with
λ x4 x5 . x0 = x4.
The subproof is completed by applying L2.