Search for blocks/addresses/...

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.
Let x2 of type ι be given.
Assume H2: x2 ∈ x0.
Let x3 of type ι be given.
Assume H3: x3 ∈ x1.
Assume H4: (λ x4 . SetAdjoin x4 (Sing 2)) x2 = (λ x4 . SetAdjoin x4 (Sing 2)) x3.
Let x4 of type ι be given.
Assume H5: x4 ∈ x2.
Claim L6: x4 ∈ (λ x5 . SetAdjoin x5 (Sing 2)) x3
Apply H4 with λ x5 x6 . x4 ∈ x5.
Apply binunionI1 with x2, Sing (Sing 2), x4.
The subproof is completed by applying H5.
Apply binunionE with x3, Sing (Sing 2), x4, x4 ∈ x3 leaving 3 subgoals.
The subproof is completed by applying L6.
Assume H7: x4 ∈ x3.
The subproof is completed by applying H7.
Assume H7: x4 ∈ Sing (Sing 2).
Apply FalseE with x4 ∈ x3.
Claim L8: x4 = Sing 2
Apply SingE with Sing 2, x4.
The subproof is completed by applying H7.
Claim L9: Sing 2 ∈ x2
Apply L8 with λ x5 x6 . x5 ∈ x2.
The subproof is completed by applying H5.
Apply H0 with False.
Let x5 of type ι be given.
Assume H10: (λ x6 . and (ordinal x6) (SNo_ x6 x0)) x5.
Apply H10 with False.
Assume H11: ordinal x5.
Assume H12: SNo_ x5 x0.
Apply H12 with False.
Assume H13: x0 ⊆ SNoElts_ x5.
Apply FalseE with (∀ x6 . x6 ∈ x5 ⟶ exactly1of2 (SetAdjoin x6 (Sing 1) ∈ x0) (x6 ∈ x0)) ⟶ False.
Apply binunionE with x5, {(λ x7 . SetAdjoin x7 (Sing 1)) x6|x6 ∈ x5}, x2, False leaving 3 subgoals.
Apply H13 with x2.
The subproof is completed by applying H2.
Assume H14: x2 ∈ x5.
Claim L15: ordinal x2
Apply ordinal_Hered with x5, x2 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H14.
Apply unknownprop_7bb148020ac74fad9e588d8f6f24c2245db7c295ea73aac9a7af2c90be710bd6.
Apply ordinal_Hered with x2, Sing 2 leaving 2 subgoals.
The subproof is completed by applying L15.
The subproof is completed by applying L9.
Assume H14: x2 ∈ {(λ x7 . SetAdjoin x7 (Sing 1)) x6|x6 ∈ x5}.
Apply ReplE_impred with x5, λ x6 . (λ x7 . SetAdjoin x7 (Sing 1)) x6, x2, False leaving 2 subgoals.
The subproof is completed by applying H14.
Let x6 of type ι be given.
Assume H15: x6 ∈ x5.
Assume H16: x2 = (λ x7 . SetAdjoin x7 (Sing 1)) x6.
Claim L17: ordinal x6
Apply ordinal_Hered with x5, x6 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H15.
Claim L18: Sing 2 ∈ (λ x7 . SetAdjoin x7 (Sing 1)) x6
Apply H16 with λ x7 x8 . Sing 2 ∈ x7.
The subproof is completed by applying L9.
Apply binunionE with x6, Sing (Sing 1), Sing 2, False leaving 3 subgoals.
The subproof is completed by applying L18.
Assume H19: Sing 2 ∈ x6.
Apply unknownprop_7bb148020ac74fad9e588d8f6f24c2245db7c295ea73aac9a7af2c90be710bd6.
Apply ordinal_Hered with x6, Sing 2 leaving 2 subgoals.
The subproof is completed by applying L17.
The subproof is completed by applying H19.
The subproof is completed by applying Sing2_notin_SingSing1.
■