Proofgold Signed Transaction

Param SNo^SNo : ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Param bij^bij : ι → ι → (ι → ι) → ο
Definition equip^equip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2
Definition finite^finite := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (equip x0 x2) ⟶ x1) ⟶ x1
Param SNoLe^SNoLe : ι → ι → ο
Definition SNo_max_of^SNo_max_of := λ x0 x1 . and (and (x1 ∈ x0) (SNo x1)) (∀ x2 . x2 ∈ x0 ⟶ SNo x2 ⟶ SNoLe x2 x1)
Param nat_p^nat_p : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param ordsucc^ordsucc : ι → ι
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 omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known bijE^bijE : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ ∀ x3 : ο . ((∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x6 = x4) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known In_0_1^In_0_1 : 0 ∈ 1
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known bijI^bijI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ bij x0 x1 x2
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem finite_max_exists^{finite_max_exists} : ∀ x0 . (∀ x1 . x1 ∈ x0 ⟶ SNo x1) ⟶ finite x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 : ο . (∀ x2 . SNo_max_of x0 x2 ⟶ x1) ⟶ x1 (proof)
Param SNo_min_of^SNo_min_of : ι → ι → ο
Param minus_SNo^minus_SNo : ι → ι
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known minus_SNo_max_min'^{minus_SNo_max_min} : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ SNo_max_of (prim5 x0 minus_SNo) x1 ⟶ SNo_min_of x0 (minus_SNo x1)
Theorem finite_min_exists^{finite_min_exists} : ∀ x0 . (∀ x1 . x1 ∈ x0 ⟶ SNo x1) ⟶ finite x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 : ο . (∀ x2 . SNo_min_of x0 x2 ⟶ x1) ⟶ x1 (proof)
Param SNoS_^SNoS_ : ι → ι
Param ordinal^ordinal : ι → ο
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_E^SNoS_E : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (SNo_ x3 x1) ⟶ x2) ⟶ x2
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Theorem ec4c7.. : SNoS_ 0 = 0 (proof)
Param add_SNo^add_SNo : ι → ι → ι
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known SNo_1^SNo_1 : SNo 1
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem add_SNo_omega_In_cases^{add_SNo_omega_In_cases} : ∀ x0 x1 . x1 ∈ omega ⟶ ∀ x2 . nat_p x2 ⟶ x0 ∈ add_SNo x1 x2 ⟶ or (x0 ∈ x1) (add_SNo x0 (minus_SNo x1) ∈ x2) (proof)
Param binintersect^binintersect : ι → ι → ι
Param SNo_extend0^SNo_extend0 : ι → ι
Param SNoElts_^SNoElts_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Param iff^iff : ο → ο → ο
Definition PNoEq_^PNoEq_ := λ x0 . λ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)
Definition SNoEq_^SNoEq_ := λ x0 x1 x2 . PNoEq_ x0 (λ x3 . x3 ∈ x1) (λ x3 . x3 ∈ x2)
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known restr_SNo^restr_SNo : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNo (binintersect x0 (SNoElts_ x1))
Known restr_SNoLev^restr_SNoLev : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoLev (binintersect x0 (SNoElts_ x1)) = x1
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Known SNoEq_tra_^SNoEq_tra_ : ∀ x0 x1 x2 x3 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x3 ⟶ SNoEq_ x0 x1 x3
Known restr_SNoEq^restr_SNoEq : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoEq_ x1 (binintersect x0 (SNoElts_ x1)) x0
Known SNo_extend0_SNoEq^{SNo_extend0_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend0 x0) x0
Known SNo_extend0_SNoLev^{SNo_extend0_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend0 x0) = ordsucc (SNoLev x0)
Known SNo_extend0_SNo^{SNo_extend0_SNo} : ∀ x0 . SNo x0 ⟶ SNo (SNo_extend0 x0)
Theorem SNo_extend0_restr_eq^{SNo_extend0_restr_eq} : ∀ x0 . SNo x0 ⟶ x0 = binintersect (SNo_extend0 x0) (SNoElts_ (SNoLev x0)) (proof)
Param SNo_extend1^SNo_extend1 : ι → ι
Known SNo_extend1_SNoEq^{SNo_extend1_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend1 x0) x0
Known SNo_extend1_SNoLev^{SNo_extend1_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend1 x0) = ordsucc (SNoLev x0)
Known SNo_extend1_SNo^{SNo_extend1_SNo} : ∀ x0 . SNo x0 ⟶ SNo (SNo_extend1 x0)
Theorem SNo_extend1_restr_eq^{SNo_extend1_restr_eq} : ∀ x0 . SNo x0 ⟶ x0 = binintersect (SNo_extend1 x0) (SNoElts_ (SNoLev x0)) (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known exp_SNo_nat_0^{exp_SNo_nat_0} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 0 = 1
Known SNo_2^SNo_2 : SNo 2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known nat_0^nat_0 : nat_p 0
Known ordinal_SNo_^ordinal_SNo_ : ∀ x0 . ordinal x0 ⟶ SNo_ x0 x0
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SNoS_E2^SNoS_E2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (SNoLev x1 ∈ x0 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNo_ (SNoLev x1) x1 ⟶ x2) ⟶ x2
Known SNo_0^SNo_0 : SNo 0
Param mul_SNo^mul_SNo : ι → ι → ι
Known exp_SNo_nat_S^{exp_SNo_nat_S} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ exp_SNo_nat x0 (ordsucc x1) = mul_SNo x0 (exp_SNo_nat x0 x1)
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Param If_i^If_i : ο → ι → ι → ι
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known exp_SNo_nat_pos^{exp_SNo_nat_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNoLt 0 (exp_SNo_nat x0 x1)
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
Known add_SNo_ordinal_ordinal^{add_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (add_SNo x0 x1)
Known iff_trans^iff_trans : ∀ x0 x1 x2 : ο . iff x0 x1 ⟶ iff x1 x2 ⟶ iff x0 x2
Known iff_sym^iff_sym : ∀ x0 x1 : ο . iff x0 x1 ⟶ iff x1 x0
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
Known omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0
Known add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known SNo_extend1_SNo_^{SNo_extend1_SNo_} : ∀ x0 . SNo x0 ⟶ SNo_ (ordsucc (SNoLev x0)) (SNo_extend1 x0)
Known SNo_extend1_In^{SNo_extend1_In} : ∀ x0 . SNo x0 ⟶ SNoLev x0 ∈ SNo_extend1 x0
Known SNo_extend0_SNo_^{SNo_extend0_SNo_} : ∀ x0 . SNo x0 ⟶ SNo_ (ordsucc (SNoLev x0)) (SNo_extend0 x0)
Known SNo_extend0_nIn^{SNo_extend0_nIn} : ∀ x0 . SNo x0 ⟶ nIn (SNoLev x0) (SNo_extend0 x0)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known restr_SNo_^restr_SNo_ : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNo_ x1 (binintersect x0 (SNoElts_ x1))
Known nat_exp_SNo_nat^{nat_exp_SNo_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_SNo_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Theorem SNoS_omega_Lev_equip^{SNoS_omega_Lev_equip} : ∀ x0 . nat_p x0 ⟶ equip {x1 ∈ SNoS_ omega|SNoLev x1 = x0} (exp_SNo_nat 2 x0) (proof)
Theorem equip_0_Empty^{equip_0_Empty} : ∀ x0 . equip x0 0 ⟶ x0 = 0 (proof)
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known ReplE'^ReplE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ x2 (x1 x3)) ⟶ ∀ x3 . x3 ∈ prim5 x0 x1 ⟶ x2 x3
Theorem finite_ind^finite_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 x2 . finite x1 ⟶ nIn x2 x1 ⟶ x0 x1 ⟶ x0 (binunion x1 (Sing x2))) ⟶ ∀ x1 . finite x1 ⟶ x0 x1 (proof)
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Theorem finite_Empty^finite_Empty : finite 0 (proof)
Theorem nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1 (proof)
Theorem a0d40.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ (∀ x2 . nat_p x2 ⟶ x1 (ordsucc x2)) ⟶ x1 x0 (proof)
Known binunion_Subq_1^{binunion_Subq_1} : ∀ x0 x1 . x0 ⊆ binunion x0 x1
Theorem adjoin_finite^{adjoin_finite} : ∀ x0 x1 . finite x0 ⟶ finite (binunion x0 (Sing x1)) (proof)
Known binunion_idr^binunion_idr : ∀ x0 . binunion x0 0 = x0
Known binunion_asso^{binunion_asso} : ∀ x0 x1 x2 . binunion x0 (binunion x1 x2) = binunion (binunion x0 x1) x2
Theorem binunion_finite^{binunion_finite} : ∀ x0 . finite x0 ⟶ ∀ x1 . finite x1 ⟶ finite (binunion x0 x1) (proof)
Param famunion^famunion : ι → (ι → ι) → ι
Known famunion_Empty^{famunion_Empty} : ∀ x0 : ι → ι . famunion 0 x0 = 0
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . 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 binunionE'^binunionE : ∀ x0 x1 x2 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3
Theorem famunion_nat_finite^{famunion_nat_finite} : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ finite (x0 x2)) ⟶ finite (famunion x1 x0) (proof)
Known SNoS_Subq^SNoS_Subq : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoS_ x0 ⊆ SNoS_ x1
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known omega_TransSet^{omega_TransSet} : TransSet omega
Theorem SNoS_finite^SNoS_finite : ∀ x0 . x0 ∈ omega ⟶ finite (SNoS_ x0) (proof)
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Param setminus^setminus : ι → ι → ι
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Theorem Subq_finite^Subq_finite : ∀ x0 . finite x0 ⟶ ∀ x1 . x1 ⊆ x0 ⟶ finite x1 (proof)
Param SNoL^SNoL : ι → ι
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Theorem SNoS_omega_SNoL_finite^{SNoS_omega_SNoL_finite} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ finite (SNoL x0) (proof)
Param SNoR^SNoR : ι → ι
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Theorem SNoS_omega_SNoR_finite^{SNoS_omega_SNoR_finite} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ finite (SNoR x0) (proof)
Theorem SNoS_omega_SNoL_max_exists^{SNoS_omega_SNoL_max_exists} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ or (SNoL x0 = 0) (∀ x1 : ο . (∀ x2 . SNo_max_of (SNoL x0) x2 ⟶ x1) ⟶ x1) (proof)
Theorem SNoS_omega_SNoR_min_exists^{SNoS_omega_SNoR_min_exists} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ or (SNoR x0 = 0) (∀ x1 : ο . (∀ x2 . SNo_min_of (SNoR x0) x2 ⟶ x1) ⟶ x1) (proof)