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Proofgold Proof

pf
Let x0 of type ι be given.
Apply Unj_eq with Inj0 x0, λ x1 x2 . x2 = x0.
Apply set_ext with {Unj x1|x1 ∈ setminus (Inj0 x0) (Sing 0)}, x0 leaving 2 subgoals.
Let x1 of type ι be given.
Assume H0: x1{Unj x2|x2 ∈ setminus (Inj0 x0) (Sing 0)}.
Apply ReplE_impred with setminus (Inj0 x0) (Sing 0), Unj, x1, x1x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Assume H1: x2setminus (Inj0 x0) (Sing 0).
Assume H2: x1 = Unj x2.
Apply setminusE with Inj0 x0, Sing 0, x2, x1x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H3: x2{Inj1 x3|x3 ∈ x0}.
Assume H4: nIn x2 (Sing 0).
Apply ReplE_impred with x0, Inj1, x2, x1x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H5: x3x0.
Assume H6: x2 = Inj1 x3.
Claim L7: x1 = x3
Apply H2 with λ x4 x5 . x5 = x3.
Apply H6 with λ x4 x5 . Unj x5 = x3.
The subproof is completed by applying Unj_Inj1_eq with x3.
Apply L7 with λ x4 x5 . x5x0.
The subproof is completed by applying H5.
Let x1 of type ι be given.
Assume H0: x1x0.
Apply Unj_Inj1_eq with x1, λ x2 x3 . x2{Unj x4|x4 ∈ setminus (Inj0 x0) (Sing 0)}.
Apply ReplI with setminus (Inj0 x0) (Sing 0), Unj, Inj1 x1.
Apply setminusI with Inj0 x0, Sing 0, Inj1 x1 leaving 2 subgoals.
Apply ReplI with x0, Inj1, x1.
The subproof is completed by applying H0.
The subproof is completed by applying Inj1NE2 with x1.