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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNoLev x0 ⊆ SNoLev x1.
Assume H3: ∀ x2 . x2 ∈ SNoLev x0 ⟶ iff (x2 ∈ x0) (x2 ∈ x1).
Apply SNoLev_ with x0, x0 ⊆ x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H4: x0 ⊆ SNoElts_ (SNoLev x0).
Assume H5: ∀ x2 . x2 ∈ SNoLev x0 ⟶ exactly1of2 ((λ x3 . SetAdjoin x3 (Sing 1)) x2 ∈ x0) (x2 ∈ x0).
Apply SNoLev_ with x1, x0 ⊆ x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H6: x1 ⊆ SNoElts_ (SNoLev x1).
Assume H7: ∀ x2 . x2 ∈ SNoLev x1 ⟶ exactly1of2 ((λ x3 . SetAdjoin x3 (Sing 1)) x2 ∈ x1) (x2 ∈ x1).
Let x2 of type ι be given.
Assume H8: x2 ∈ x0.
Claim L9: ...
...
Apply binunionE with SNoLev x0, {(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ SNoLev x0}, x2, x2 ∈ x1 leaving 3 subgoals.
The subproof is completed by applying L9.
Assume H10: x2 ∈ SNoLev x0.
Apply H3 with x2, x2 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H10.
Assume H11: x2 ∈ x0 ⟶ x2 ∈ x1.
Assume H12: x2 ∈ x1 ⟶ x2 ∈ x0.
Apply H11.
The subproof is completed by applying H8.
Assume H10: x2 ∈ {(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ SNoLev x0}.
Apply ReplE_impred with SNoLev x0, λ x3 . (λ x4 . SetAdjoin x4 (Sing 1)) x3, x2, x2 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H10.
Let x3 of type ι be given.
Assume H11: x3 ∈ SNoLev x0.
Assume H12: x2 = (λ x4 . SetAdjoin x4 (Sing 1)) x3.
Claim L13: x3 ∈ SNoLev x1
Apply H2 with ....
...
Apply exactly1of2_E with (λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x1, x3 ∈ x1, x2 ∈ x1 leaving 3 subgoals.
Apply H7 with x3.
The subproof is completed by applying L13.
Assume H14: (λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x1.
Assume H15: nIn x3 x1.
Apply H12 with λ x4 x5 . x5 ∈ x1.
The subproof is completed by applying H14.
Assume H14: nIn ((λ x4 . SetAdjoin x4 (Sing 1)) x3) x1.
Assume H15: x3 ∈ x1.
Apply FalseE with x2 ∈ x1.
Apply exactly1of2_E with (λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x0, x3 ∈ x0, False leaving 3 subgoals.
Apply H5 with x3.
The subproof is completed by applying H11.
Assume H16: (λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x0.
Assume H17: nIn x3 x0.
Apply H17.
Apply H3 with x3, x3 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H11.
Assume H18: x3 ∈ x0 ⟶ x3 ∈ x1.
Assume H19: x3 ∈ x1 ⟶ x3 ∈ x0.
Apply H19.
The subproof is completed by applying H15.
Assume H16: nIn ((λ x4 . SetAdjoin x4 (Sing 1)) x3) x0.
Assume H17: x3 ∈ x0.
Apply H16.
Apply H12 with λ x4 x5 . x4 ∈ x0.
The subproof is completed by applying H8.
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