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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιι be given.
Apply set_ext with {x1 x2|x2 ∈ x0}, famunion x0 (λ x2 . Sing (x1 x2)) leaving 2 subgoals.
Let x2 of type ι be given.
Assume H0: x2{x1 x3|x3 ∈ x0}.
Apply ReplE_impred with x0, x1, x2, x2famunion x0 (λ x3 . Sing (x1 x3)) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: x3x0.
Assume H2: x2 = x1 x3.
Apply famunionI with x0, λ x4 . Sing (x1 x4), x3, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H2 with λ x4 x5 . x2Sing x4.
The subproof is completed by applying SingI with x2.
Let x2 of type ι be given.
Assume H0: x2famunion x0 (λ x3 . Sing (x1 x3)).
Apply famunionE with x0, λ x3 . Sing (x1 x3), x2, x2{x1 x3|x3 ∈ x0} leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: and (x3x0) (x2Sing (x1 x3)).
Apply H1 with x2{x1 x4|x4 ∈ x0}.
Assume H2: x3x0.
Assume H3: x2Sing (x1 x3).
Claim L4: x2 = x1 x3
Apply SingE with x1 x3, x2.
The subproof is completed by applying H3.
Apply L4 with λ x4 x5 . x5{x1 x6|x6 ∈ x0}.
Apply ReplI with x0, x1, x3.
The subproof is completed by applying H2.