Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0:
x1 ∈ {Unj x2|x2 ∈ x0,∃ x3 . Inj0 x3 = x2}.
Apply Inj0_pair_0_eq with
λ x2 x3 : ι → ι . x2 x1 ∈ x0.
Apply ReplSepE_impred with
x0,
λ x2 . ∃ x3 . Inj0 x3 = x2,
Unj,
x1,
Inj0 x1 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Assume H2:
∃ x3 . Inj0 x3 = x2.
Apply H2 with
Inj0 x1 ∈ x0.
Let x3 of type ι be given.
Apply H3 with
λ x4 x5 . Inj0 x5 ∈ x0.
Apply H4 with
λ x4 x5 . Inj0 (Unj x4) ∈ x0.
Apply Unj_Inj0_eq with
x3,
λ x4 x5 . Inj0 x5 ∈ x0.
Apply H4 with
λ x4 x5 . x5 ∈ x0.
The subproof is completed by applying H1.