Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param ordsucc^ordsucc : ι → ι
Param Sing^Sing : ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem Subq_1_Sing0^Subq_1_Sing0 : 1 ⊆ Sing 0 (proof)
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem Subq_Sing0_1^Subq_Sing0_1 : Sing 0 ⊆ 1 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Theorem eq_1_Sing0^eq_1_Sing0 : 1 = Sing 0 (proof)
Param UPair^UPair : ι → ι → ι
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Theorem Subq_2_UPair01^{Subq_2_UPair01} : 2 ⊆ UPair 0 1 (proof)
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known In_0_2^In_0_2 : 0 ∈ 2
Known In_1_2^In_1_2 : 1 ∈ 2
Theorem Subq_UPair01_2^{Subq_UPair01_2} : UPair 0 1 ⊆ 2 (proof)
Theorem eq_2_UPair01^eq_2_UPair01 : 2 = UPair 0 1 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Definition ordinal^ordinal := λ x0 . and (TransSet x0) (∀ x1 . x1 ∈ x0 ⟶ TransSet x1)
Known In_ind^In_ind : ∀ x0 : ι → ο . (∀ x1 . (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . x0 x1
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem ordinal_ind^ordinal_ind : ∀ x0 : ι → ο . (∀ x1 . ordinal x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . ordinal x1 ⟶ x0 x1 (proof)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem least_ordinal_ex^{least_ordinal_ex} : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . and (ordinal x2) (x0 x2) ⟶ x1) ⟶ x1) ⟶ ∀ x1 : ο . (∀ x2 . and (and (ordinal x2) (x0 x2)) (∀ x3 . x3 ∈ x2 ⟶ not (x0 x3)) ⟶ x1) ⟶ x1 (proof)
Param nat_p^nat_p : ι → ο
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_1^nat_1 : nat_p 1
Known ordinal_1^ordinal_1 : ordinal 1
Known nat_2^nat_2 : nat_p 2
Known ordinal_2^ordinal_2 : ordinal 2
Param omega^omega : ι
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known omega_TransSet^{omega_TransSet} : TransSet omega
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known TransSet_ordsucc_In_Subq^{TransSet_ordsucc_In_Subq} : ∀ x0 . TransSet x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known or3I1^or3I1 : ∀ x0 x1 x2 : ο . x0 ⟶ or (or x0 x1) x2
Known or3I3^or3I3 : ∀ x0 x1 x2 : ο . x2 ⟶ or (or x0 x1) x2
Known or3I2^or3I2 : ∀ x0 x1 x2 : ο . x1 ⟶ or (or x0 x1) x2
Known or3E^or3E : ∀ x0 x1 x2 : ο . or (or x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x3) ⟶ (x1 ⟶ x3) ⟶ (x2 ⟶ x3) ⟶ x3
Known ordinal_trichotomy_or^{ordinal_trichotomy_or} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (or (x0 ∈ x1) (x0 = x1)) (x1 ∈ x0)
Param exactly1of3^exactly1of3 : ο → ο → ο → ο
Known exactly1of3_I1^{exactly1of3_I1} : ∀ x0 x1 x2 : ο . x0 ⟶ not x1 ⟶ not x2 ⟶ exactly1of3 x0 x1 x2
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known exactly1of3_I2^{exactly1of3_I2} : ∀ x0 x1 x2 : ο . not x0 ⟶ x1 ⟶ not x2 ⟶ exactly1of3 x0 x1 x2
Known exactly1of3_I3^{exactly1of3_I3} : ∀ x0 x1 x2 : ο . not x0 ⟶ not x1 ⟶ x2 ⟶ exactly1of3 x0 x1 x2
Theorem ordinal_trichotomy^{ordinal_trichotomy} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ exactly1of3 (x0 ∈ x1) (x0 = x1) (x1 ∈ x0) (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known ordinal_linear^{ordinal_linear} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ⊆ x1) (x1 ⊆ x0)
Known ordinal_ordsucc_In_eq^{ordinal_ordsucc_In_eq} : ∀ x0 x1 . ordinal x0 ⟶ x1 ∈ x0 ⟶ or (ordsucc x1 ∈ x0) (x0 = ordsucc x1)
Known ordinal_lim_or_succ^{ordinal_lim_or_succ} : ∀ x0 . ordinal x0 ⟶ or (∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ x0) (∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known UnionE_impred^{UnionE_impred} : ∀ x0 x1 . x1 ∈ prim3 x0 ⟶ ∀ x2 : ο . (∀ x3 . x1 ∈ x3 ⟶ x3 ∈ x0 ⟶ x2) ⟶ x2
Known UnionI^UnionI : ∀ x0 x1 x2 . x1 ∈ x2 ⟶ x2 ∈ x0 ⟶ x1 ∈ prim3 x0
Known ordinal_Union^{ordinal_Union} : ∀ x0 . (∀ x1 . x1 ∈ x0 ⟶ ordinal x1) ⟶ ordinal (prim3 x0)
Param famunion^famunion : ι → (ι → ι) → ι
Known famunionE^famunionE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 ∈ x1 x4) ⟶ x3) ⟶ x3
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known ordinal_famunion^{ordinal_famunion} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ordinal (x1 x2)) ⟶ ordinal (famunion x0 x1)
Param binintersect^binintersect : ι → ι → ι
Known binintersect_Subq_eq_1^{binintersect_Subq_eq_1} : ∀ x0 x1 . x0 ⊆ x1 ⟶ binintersect x0 x1 = x0
Known binintersect_com^{binintersect_com} : ∀ x0 x1 . binintersect x0 x1 = binintersect x1 x0
Known ordinal_binintersect^{ordinal_binintersect} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal (binintersect x0 x1)
Param binunion^binunion : ι → ι → ι
Known Subq_binunion_eq^{Subq_binunion_eq} : ∀ x0 x1 . x0 ⊆ x1 = (binunion x0 x1 = x1)
Known binunion_com^binunion_com : ∀ x0 x1 . binunion x0 x1 = binunion x1 x0
Known ordinal_binunion^{ordinal_binunion} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal (binunion x0 x1)
Param Sep^Sep : ι → (ι → ο) → ι
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known ordinal_Sep^ordinal_Sep : ∀ x0 . ordinal x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x2 ⟶ x1 x2 ⟶ x1 x3) ⟶ ordinal (Sep x0 x1)
Param In_rec_i^In_rec_i : (ι → (ι → ι) → ι) → ι → ι
Definition Inj1^Inj1 := In_rec_i (λ x0 . λ x1 : ι → ι . binunion (Sing 0) (prim5 x0 x1))
Known In_rec_i_eq^In_rec_i_eq : ∀ x0 : ι → (ι → ι) → ι . (∀ x1 . ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . In_rec_i x0 x1 = x0 x1 (In_rec_i x0)
Known ReplEq_ext^ReplEq_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ prim5 x0 x1 = prim5 x0 x2
Theorem Inj1_eq^Inj1_eq : ∀ x0 . Inj1 x0 = binunion (Sing 0) (prim5 x0 Inj1) (proof)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Theorem Inj1I1^Inj1I1 : ∀ x0 . 0 ∈ Inj1 x0 (proof)
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem Inj1I2^Inj1I2 : ∀ x0 x1 . x1 ∈ x0 ⟶ Inj1 x1 ∈ Inj1 x0 (proof)
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known ReplE^ReplE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = x1 x4) ⟶ x3) ⟶ x3
Theorem Inj1E^Inj1E : ∀ x0 x1 . x1 ∈ Inj1 x0 ⟶ or (x1 = 0) (∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (x1 = Inj1 x3) ⟶ x2) ⟶ x2) (proof)
Theorem Inj1NE1^Inj1NE1 : ∀ x0 . Inj1 x0 = 0 ⟶ ∀ x1 : ο . x1 (proof)
Theorem Inj1NE2^Inj1NE2 : ∀ x0 . nIn (Inj1 x0) (Sing 0) (proof)
Definition Inj0^Inj0 := λ x0 . prim5 x0 Inj1
Theorem Inj0I^Inj0I : ∀ x0 x1 . x1 ∈ x0 ⟶ Inj1 x1 ∈ Inj0 x0 (proof)
Theorem Inj0E^Inj0E : ∀ x0 x1 . x1 ∈ Inj0 x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (x1 = Inj1 x3) ⟶ x2) ⟶ x2 (proof)
Param setminus^setminus : ι → ι → ι
Definition Unj^Unj := In_rec_i (λ x0 . prim5 (setminus x0 (Sing 0)))
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem Unj_eq^Unj_eq : ∀ x0 . Unj x0 = prim5 (setminus x0 (Sing 0)) Unj (proof)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known exandE_i^exandE_i : ∀ x0 x1 : ι → ο . (∀ x2 : ο . (∀ x3 . and (x0 x3) (x1 x3) ⟶ x2) ⟶ x2) ⟶ ∀ x2 : ο . (∀ x3 . x0 x3 ⟶ x1 x3 ⟶ x2) ⟶ x2
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Theorem Unj_Inj1_eq^Unj_Inj1_eq : ∀ x0 . Unj (Inj1 x0) = x0 (proof)
Theorem Inj1_inj^Inj1_inj : ∀ x0 x1 . Inj1 x0 = Inj1 x1 ⟶ x0 = x1 (proof)
Theorem Unj_Inj0_eq^Unj_Inj0_eq : ∀ x0 . Unj (Inj0 x0) = x0 (proof)
Theorem Inj0_inj^Inj0_inj : ∀ x0 x1 . Inj0 x0 = Inj0 x1 ⟶ x0 = x1 (proof)
Known Repl_Empty^Repl_Empty : ∀ x0 : ι → ι . prim5 0 x0 = 0
Theorem Inj0_0^Inj0_0 : Inj0 0 = 0 (proof)
Known andER^andER : ∀ x0 x1 : ο . and x0 x1 ⟶ x1
Theorem Inj0_Inj1_neq^{Inj0_Inj1_neq} : ∀ x0 x1 . Inj0 x0 = Inj1 x1 ⟶ ∀ x2 : ο . x2 (proof)
Definition setsum^setsum := λ x0 x1 . binunion (prim5 x0 Inj0) (prim5 x1 Inj1)
Theorem Inj0_setsum^Inj0_setsum : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ Inj0 x2 ∈ setsum x0 x1 (proof)
Theorem Inj1_setsum^Inj1_setsum : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ Inj1 x2 ∈ setsum x0 x1 (proof)
Theorem setsum_Inj_inv^{setsum_Inj_inv} : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = Inj0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = Inj1 x4) ⟶ x3) ⟶ x3) (proof)
Theorem Inj0_setsum_0L^{Inj0_setsum_0L} : ∀ x0 . setsum 0 x0 = Inj0 x0 (proof)
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem Inj1_setsum_1L^{Inj1_setsum_1L} : ∀ x0 . setsum 1 x0 = Inj1 x0 (proof)
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Theorem nat_setsum_ordsucc^{nat_setsum1_ordsucc} : ∀ x0 . nat_p x0 ⟶ setsum 1 x0 = ordsucc x0 (proof)
Theorem setsum_0_0^setsum_0_0 : setsum 0 0 = 0 (proof)
Known nat_0^nat_0 : nat_p 0
Theorem setsum_1_0_1^setsum_1_0_1 : setsum 1 0 = 1 (proof)
Theorem setsum_1_1_2^setsum_1_1_2 : setsum 1 1 = 2 (proof)
Theorem pairSubq^pairSubq : ∀ x0 x1 x2 x3 . x0 ⊆ x2 ⟶ x1 ⊆ x3 ⟶ setsum x0 x1 ⊆ setsum x2 x3 (proof)
Param If_i^If_i : ο → ι → ι → ι
Definition combine_funcs^{combine_funcs} := λ x0 x1 . λ x2 x3 : ι → ι . λ x4 . If_i (x4 = Inj0 (Unj x4)) (x2 (Unj x4)) (x3 (Unj x4))
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem combine_funcs_eq1^{combine_funcs_eq1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj0 x4) = x2 x4 (proof)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem combine_funcs_eq2^{combine_funcs_eq2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj1 x4) = x3 x4 (proof)
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Definition proj0^proj0 := λ x0 . ReplSep x0 (λ x1 . ∀ x2 : ο . (∀ x3 . Inj0 x3 = x1 ⟶ x2) ⟶ x2) Unj
Definition proj1^proj1 := λ x0 . ReplSep x0 (λ x1 . ∀ x2 : ο . (∀ x3 . Inj1 x3 = x1 ⟶ x2) ⟶ x2) Unj
Theorem Inj0_pair_0_eq^{Inj0_pair_0_eq} : Inj0 = setsum 0 (proof)
Theorem Inj1_pair_1_eq^{Inj1_pair_1_eq} : Inj1 = setsum 1 (proof)
Theorem pairI0^pairI0 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ setsum 0 x2 ∈ setsum x0 x1 (proof)
Theorem pairI1^pairI1 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ setsum 1 x2 ∈ setsum x0 x1 (proof)
Theorem pairE^pairE : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = setsum 0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = setsum 1 x4) ⟶ x3) ⟶ x3) (proof)
Theorem pairE0^pairE0 : ∀ x0 x1 x2 . setsum 0 x2 ∈ setsum x0 x1 ⟶ x2 ∈ x0 (proof)
Theorem pairE1^pairE1 : ∀ x0 x1 x2 . setsum 1 x2 ∈ setsum x0 x1 ⟶ x2 ∈ x1 (proof)
Param iff^iff : ο → ο → ο
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem pairEq^pairEq : ∀ x0 x1 x2 . iff (x2 ∈ setsum x0 x1) (or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = setsum 0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = setsum 1 x4) ⟶ x3) ⟶ x3)) (proof)
Known ReplSepI^ReplSepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⟶ x2 x3 ∈ ReplSep x0 x1 x2
Theorem proj0I^proj0I : ∀ x0 x1 . setsum 0 x1 ∈ x0 ⟶ x1 ∈ proj0 x0 (proof)
Known ReplSepE_impred^{ReplSepE_impred} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ ReplSep x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4
Theorem proj0E^proj0E : ∀ x0 x1 . x1 ∈ proj0 x0 ⟶ setsum 0 x1 ∈ x0 (proof)
Theorem proj1I^proj1I : ∀ x0 x1 . setsum 1 x1 ∈ x0 ⟶ x1 ∈ proj1 x0 (proof)
Theorem proj1E^proj1E : ∀ x0 x1 . x1 ∈ proj1 x0 ⟶ setsum 1 x1 ∈ x0 (proof)
Theorem proj0_pair_eq^{proj0_pair_eq} : ∀ x0 x1 . proj0 (setsum x0 x1) = x0 (proof)
Theorem proj1_pair_eq^{proj1_pair_eq} : ∀ x0 x1 . proj1 (setsum x0 x1) = x1 (proof)
Theorem pair_inj^pair_inj : ∀ x0 x1 x2 x3 . setsum x0 x1 = setsum x2 x3 ⟶ and (x0 = x2) (x1 = x3) (proof)
Theorem pair_eta_Subq_proj^{pair_eta_Subq_proj} : ∀ x0 . setsum (proj0 x0) (proj1 x0) ⊆ x0 (proof)