TPTP Problem File: SLH0963^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0022_Restrict_Bounds/prob_00028_000783__17590072_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1485 ( 623 unt; 198 typ; 0 def)
% Number of atoms : 4325 (1724 equ; 0 cnn)
% Maximal formula atoms : 45 ( 3 avg)
% Number of connectives : 13165 ( 840 ~; 121 |; 372 &;9866 @)
% ( 0 <=>;1966 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 6 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 739 ( 739 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 13 con; 0-4 aty)
% Number of variables : 4016 ( 290 ^;3526 !; 200 ?;4016 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:37.471
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
produc1132964494702330949_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Relational____Calculus__Oterm_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
set_se6865892389300016395la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
set_Re381260168593705685la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Oterm_Itf__a_J_J,type,
list_R6823256787227418703term_a: $tType ).
thf(ty_n_t__Set__Oset_It__Relational____Calculus__Oterm_Itf__a_J_J,type,
set_Re5178783185447174953term_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Oterm_It__Nat__Onat_J,type,
relational_term_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relational_fmla_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Relational____Calculus__Oterm_Itf__a_J,type,
relational_term_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
finite5238674622262875500la_a_b: set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
fun_up6290740034186280484la_a_b: ( nat > set_Re381260168593705685la_a_b ) > nat > set_Re381260168593705685la_a_b > nat > set_Re381260168593705685la_a_b ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
minus_9215201808853403479_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
sup_su1471977682094119364_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
sup_su5130108678486352897la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
sup_su4783144482993978935la_a_b: set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
lattic5380700691367270794_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
lattic7150925611526040158la_a_b: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_List_Olist_Omap_001t__Relational____Calculus__Oterm_Itf__a_J_001t__Relational____Calculus__Oterm_Itf__a_J,type,
map_Re5736185711816362116term_a: ( relational_term_a > relational_term_a ) > list_R6823256787227418703term_a > list_R6823256787227418703term_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
set_Re9104216502384355786la_a_b: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_List_Olist_Oset_001t__Relational____Calculus__Oterm_Itf__a_J,type,
set_Re3569617851344498910term_a: list_R6823256787227418703term_a > set_Re5178783185447174953term_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
bot_bo8852203127187332700_a_b_o: relational_fmla_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
bot_bo4495933725496725865la_a_b: set_Re381260168593705685la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
bot_bo2891247006866115487la_a_b: set_se6865892389300016395la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
ord_le6021219098528097948_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
ord_le1577343677690852715la_a_b: set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_001_062_It__Nat__Onat_Mtf__a_J,type,
produc2895298938842563487_nat_a: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > produc5835360497134304175_nat_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
produc4282057684358614024_b_nat: relational_fmla_a_b > nat > produc7366699395886430672_b_nat ).
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thf(sy_c_Relational__Calculus_Oadom_001tf__b_001tf__a,type,
relational_adom_b_a: ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Relational__Calculus_Oap_001tf__a_001tf__b,type,
relational_ap_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_001tf__a_001tf__b,type,
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thf(sy_c_Relational__Calculus_Ocov_H_001tf__a_001tf__b,type,
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thf(sy_c_Relational__Calculus_Ocp_001tf__a_001tf__b,type,
relational_cp_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ocpropagated_001tf__a_001tf__b,type,
relati1591879772219623554ed_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts_001tf__a_001tf__b,type,
relational_csts_a_b: relational_fmla_a_b > set_a ).
thf(sy_c_Relational__Calculus_Ocsts__rel_001tf__a_001tf__b,type,
relati7137348651719826542el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts__term_001t__Nat__Onat,type,
relati694035416245573993rm_nat: relational_term_nat > set_nat ).
thf(sy_c_Relational__Calculus_Ocsts__term_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relati1926769566493843000la_a_b: relati112041753218324778la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Relational__Calculus_Ocsts__term_001tf__a,type,
relati6638259395341848997term_a: relational_term_a > set_a ).
thf(sy_c_Relational__Calculus_Oeqs_001tf__a_001tf__b,type,
relational_eqs_a_b: nat > set_Re381260168593705685la_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oequiv_001tf__a_001tf__b,type,
relational_equiv_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oerase_001tf__a_001tf__b,type,
relational_erase_a_b: relational_fmla_a_b > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oerase__rel_001tf__a_001tf__b,type,
relati5987653437628155313el_a_b: produc7366699395886430672_b_nat > produc7366699395886430672_b_nat > $o ).
thf(sy_c_Relational__Calculus_Oeval_001tf__a_001tf__b,type,
relational_eval_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oeval__on_001tf__a_001tf__b,type,
relati8814510239606734169on_a_b: set_nat > relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oexists_001tf__a_001tf__b,type,
relati3989891337220013914ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OBool_001tf__a_001tf__b,type,
relational_Bool_a_b: $o > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OConj_001tf__a_001tf__b,type,
relational_Conj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ODisj_001tf__a_001tf__b,type,
relational_Disj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OEq_001tf__a_001tf__b,type,
relational_Eq_a_b: nat > relational_term_a > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OExists_001tf__a_001tf__b,type,
relati591517084277583526ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ONeg_001tf__a_001tf__b,type,
relational_Neg_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OPred_001tf__b_001tf__a,type,
relational_Pred_b_a: b > list_R6823256787227418703term_a > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_Oset2__fmla_001tf__a_001tf__b,type,
relati8924981150291758614la_a_b: relational_fmla_a_b > set_b ).
thf(sy_c_Relational__Calculus_Ofresh2_001tf__a_001tf__b,type,
relati2677767559083392098h2_a_b: nat > nat > relational_fmla_a_b > nat ).
thf(sy_c_Relational__Calculus_Ofv_001tf__a_001tf__b,type,
relational_fv_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Ofv__rel_001tf__a_001tf__b,type,
relati5703530512245835757el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ofv__term__set_001tf__a,type,
relati6004689760767320788_set_a: relational_term_a > set_nat ).
thf(sy_c_Relational__Calculus_Ofv__terms__set_001tf__a,type,
relati4569515538964159125_set_a: list_R6823256787227418703term_a > set_nat ).
thf(sy_c_Relational__Calculus_Ogen_001tf__a_001tf__b,type,
relational_gen_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogen_H_001tf__a_001tf__b,type,
relational_gen_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogenempty_001tf__a_001tf__b,type,
relati5999705594545617851ty_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onocp_001tf__a_001tf__b,type,
relational_nocp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onocp__rel_001tf__a_001tf__b,type,
relati3149960101488570543el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onongens_001tf__a_001tf__b,type,
relati62690040636126068ns_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oqp_001tf__a_001tf__b,type,
relational_qp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqp__impl_001tf__a_001tf__b,type,
relati3725921752842749053pl_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqp__impl__rel_001tf__a_001tf__b,type,
relati7364465619720499582el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqps_001tf__a_001tf__b,type,
relational_qps_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Relational__Calculus_Orrb_001tf__a_001tf__b,type,
relational_rrb_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osat_001tf__a_001tf__b,type,
relational_sat_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o ).
thf(sy_c_Relational__Calculus_Osr_001tf__a_001tf__b,type,
relational_sr_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osub_001tf__a_001tf__b,type,
relational_sub_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Relational__Calculus_Osub__rel_001tf__a_001tf__b,type,
relati7309537865537208983el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osubst_001tf__a_001tf__b,type,
relational_subst_a_b: relational_fmla_a_b > nat > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Osubst__term_001tf__a,type,
relati7175845559408349773term_a: relational_term_a > nat > nat > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_OConst_001tf__a,type,
relational_Const_a: a > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_OVar_001tf__a,type,
relational_Var_a: nat > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_Ocase__term_001tf__a_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relati582353067970734056la_a_b: ( a > relational_fmla_a_b ) > ( nat > relational_fmla_a_b ) > relational_term_a > relational_fmla_a_b ).
thf(sy_c_Restrict__Bounds_Oflat__Disj_001tf__a_001tf__b,type,
restri569617705344514291sj_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Restrict__Bounds_Oflat__Disj__rel_001tf__a_001tf__b,type,
restri7773364413411414152el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
collec3419995626248312948la_a_b: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
collec2099942116761351594la_a_b: ( set_Re381260168593705685la_a_b > $o ) > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_4386371547000553590la_a_b: ( nat > relational_fmla_a_b ) > set_nat > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_654480401538864556la_a_b: ( nat > set_Re381260168593705685la_a_b ) > set_nat > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
image_341122591648980342_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > set_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_8719518604786020652et_nat: ( relational_fmla_a_b > set_nat ) > set_Re381260168593705685la_a_b > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_8209480069293074043la_a_b: ( relational_fmla_a_b > set_Re381260168593705685la_a_b ) > set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Oterm_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_1080223610614215258_set_a: ( relational_term_a > set_a ) > set_Re5178783185447174953term_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert2023870700798818565la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
is_sin1114528679824004833la_a_b: set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
remove4261432235257513082la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
the_el7486773796875720352la_a_b: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
accp_P4351966040938400857_b_nat: ( produc7366699395886430672_b_nat > produc7366699395886430672_b_nat > $o ) > produc7366699395886430672_b_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
accp_R989495437599811158la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_Q_H,type,
q2: relational_fmla_a_b ).
% Relevant facts (1277)
thf(fact_0_finite__flat__Disj,axiom,
! [Q: relational_fmla_a_b] : ( finite5600759454172676150la_a_b @ ( restri569617705344514291sj_a_b @ Q ) ) ).
% finite_flat_Disj
thf(fact_1_fv__flat__DisjD,axiom,
! [Q2: relational_fmla_a_b,Q: relational_fmla_a_b,X: nat] :
( ( member4680049679412964150la_a_b @ Q2 @ ( restri569617705344514291sj_a_b @ Q ) )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
=> ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).
% fv_flat_DisjD
thf(fact_2_cpropagated__simps_I1_J,axiom,
! [B: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B ) ) ).
% cpropagated_simps(1)
thf(fact_3_cpropagated__simps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P @ Ts ) ) ).
% cpropagated_simps(2)
thf(fact_4_cpropagated__sub,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member4680049679412964150la_a_b @ Q2 @ ( relational_sub_a_b @ Q ) )
=> ( relati1591879772219623554ed_a_b @ Q2 ) ) ) ).
% cpropagated_sub
thf(fact_5_nocp__cpropagated,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ Q )
=> ( relati1591879772219623554ed_a_b @ Q ) ) ).
% nocp_cpropagated
thf(fact_6_cpropagated__cp,axiom,
! [Q: relational_fmla_a_b] : ( relati1591879772219623554ed_a_b @ ( relational_cp_a_b @ Q ) ) ).
% cpropagated_cp
thf(fact_7_cpropagated__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% cpropagated_cp_triv
thf(fact_8_flat__Disj_Osimps_I1_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q1 ) @ ( restri569617705344514291sj_a_b @ Q22 ) ) ) ).
% flat_Disj.simps(1)
thf(fact_9_cpropagated__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q22 ) ) ) ).
% cpropagated_simps(6)
thf(fact_10_cpropagated__simps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% cpropagated_simps(4)
thf(fact_11_fmla_Oinject_I6_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
( ( ( relational_Disj_a_b @ X61 @ X62 )
= ( relational_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fmla.inject(6)
thf(fact_12_fmla_Oinject_I4_J,axiom,
! [X4: relational_fmla_a_b,Y4: relational_fmla_a_b] :
( ( ( relational_Neg_a_b @ X4 )
= ( relational_Neg_a_b @ Y4 ) )
= ( X4 = Y4 ) ) ).
% fmla.inject(4)
thf(fact_13_fmla_Oinject_I2_J,axiom,
! [X2: $o,Y2: $o] :
( ( ( relational_Bool_a_b @ X2 )
= ( relational_Bool_a_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% fmla.inject(2)
thf(fact_14_fmla_Oinject_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
( ( ( relational_Pred_b_a @ X11 @ X12 )
= ( relational_Pred_b_a @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fmla.inject(1)
thf(fact_15_cp__idem,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relational_cp_a_b @ Q ) )
= ( relational_cp_a_b @ Q ) ) ).
% cp_idem
thf(fact_16_equiv__refl,axiom,
! [Q: relational_fmla_a_b] : ( relational_equiv_a_b @ Q @ Q ) ).
% equiv_refl
thf(fact_17_cp_Osimps_I7_J,axiom,
! [V: $o] :
( ( relational_cp_a_b @ ( relational_Bool_a_b @ V ) )
= ( relational_Bool_a_b @ V ) ) ).
% cp.simps(7)
thf(fact_18_cp_Osimps_I6_J,axiom,
! [V: b,Va: list_R6823256787227418703term_a] :
( ( relational_cp_a_b @ ( relational_Pred_b_a @ V @ Va ) )
= ( relational_Pred_b_a @ V @ Va ) ) ).
% cp.simps(6)
thf(fact_19_fv_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
= ( relational_fv_a_b @ Phi ) ) ).
% fv.simps(4)
thf(fact_20_nocp_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q22 ) ) ) ).
% nocp.simps(6)
thf(fact_21_nocp_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% nocp.simps(4)
thf(fact_22_nocp_Osimps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] : ( relational_nocp_a_b @ ( relational_Pred_b_a @ P @ Ts ) ) ).
% nocp.simps(2)
thf(fact_23_nocp_Osimps_I1_J,axiom,
! [B: $o] :
~ ( relational_nocp_a_b @ ( relational_Bool_a_b @ B ) ) ).
% nocp.simps(1)
thf(fact_24_fmla_Odistinct_I33_J,axiom,
! [X4: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(33)
thf(fact_25_fmla_Odistinct_I19_J,axiom,
! [X2: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(19)
thf(fact_26_fmla_Odistinct_I15_J,axiom,
! [X2: $o,X4: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(15)
thf(fact_27_fmla_Odistinct_I9_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(9)
thf(fact_28_fmla_Odistinct_I5_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X4: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(5)
thf(fact_29_fmla_Odistinct_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X2: $o] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Bool_a_b @ X2 ) ) ).
% fmla.distinct(1)
thf(fact_30_equiv__cp,axiom,
! [Q: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_cp_a_b @ Q ) @ Q ) ).
% equiv_cp
thf(fact_31_equiv__sym,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q22 )
=> ( relational_equiv_a_b @ Q22 @ Q1 ) ) ).
% equiv_sym
thf(fact_32_equiv__trans,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q22 )
=> ( ( relational_equiv_a_b @ Q22 @ Q3 )
=> ( relational_equiv_a_b @ Q1 @ Q3 ) ) ) ).
% equiv_trans
thf(fact_33_nocp__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% nocp_cp_triv
thf(fact_34_equiv__cp__cong,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q @ Q2 )
=> ( relational_equiv_a_b @ ( relational_cp_a_b @ Q ) @ ( relational_cp_a_b @ Q2 ) ) ) ).
% equiv_cp_cong
thf(fact_35_equiv__Neg__cong,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q @ Q2 )
=> ( relational_equiv_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_Neg_a_b @ Q2 ) ) ) ).
% equiv_Neg_cong
thf(fact_36_equiv__Disj__cong,axiom,
! [Q1: relational_fmla_a_b,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q12 )
=> ( ( relational_equiv_a_b @ Q22 @ Q23 )
=> ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) ) ) ).
% equiv_Disj_cong
thf(fact_37_cpropagated__nocp,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_nocp
thf(fact_38_equiv__Disj__Assoc,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,Q3: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ Q3 ) @ ( relational_Disj_a_b @ Q1 @ ( relational_Disj_a_b @ Q22 @ Q3 ) ) ) ).
% equiv_Disj_Assoc
thf(fact_39_finite__Un,axiom,
! [F: set_list_a,G: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G ) )
= ( ( finite_finite_list_a @ F )
& ( finite_finite_list_a @ G ) ) ) ).
% finite_Un
thf(fact_40_finite__Un,axiom,
! [F: set_se6865892389300016395la_a_b,G: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ ( sup_su4783144482993978935la_a_b @ F @ G ) )
= ( ( finite5238674622262875500la_a_b @ F )
& ( finite5238674622262875500la_a_b @ G ) ) ) ).
% finite_Un
thf(fact_41_finite__Un,axiom,
! [F: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) )
= ( ( finite1152437895449049373et_nat @ F )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_42_finite__Un,axiom,
! [F: set_b,G: set_b] :
( ( finite_finite_b @ ( sup_sup_set_b @ F @ G ) )
= ( ( finite_finite_b @ F )
& ( finite_finite_b @ G ) ) ) ).
% finite_Un
thf(fact_43_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_44_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_45_finite__Un,axiom,
! [F: set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F @ G ) )
= ( ( finite5600759454172676150la_a_b @ F )
& ( finite5600759454172676150la_a_b @ G ) ) ) ).
% finite_Un
thf(fact_46_UnCI,axiom,
! [C: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_47_UnCI,axiom,
! [C: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ C @ B2 )
=> ( member3481406638322139244la_a_b @ C @ A ) )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ).
% UnCI
thf(fact_48_UnCI,axiom,
! [C: b,B2: set_b,A: set_b] :
( ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ A ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% UnCI
thf(fact_49_UnCI,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_50_UnCI,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_51_UnCI,axiom,
! [C: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ C @ B2 )
=> ( member4680049679412964150la_a_b @ C @ A ) )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% UnCI
thf(fact_52_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_53_Un__iff,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( ( member3481406638322139244la_a_b @ C @ A )
| ( member3481406638322139244la_a_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_54_Un__iff,axiom,
! [C: b,A: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B2 ) )
= ( ( member_b @ C @ A )
| ( member_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_55_Un__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_56_Un__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_57_Un__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
| ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_58_mem__Collect__eq,axiom,
! [A2: b,P2: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ A2 @ ( collec2099942116761351594la_a_b @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b,P2: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( collec2099942116761351594la_a_b
@ ^ [X3: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_70_Collect__cong,axiom,
! [P2: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ! [X5: relational_fmla_a_b] :
( ( P2 @ X5 )
= ( Q @ X5 ) )
=> ( ( collec3419995626248312948la_a_b @ P2 )
= ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P2: set_Re381260168593705685la_a_b > $o,Q: set_Re381260168593705685la_a_b > $o] :
( ! [X5: set_Re381260168593705685la_a_b] :
( ( P2 @ X5 )
= ( Q @ X5 ) )
=> ( ( collec2099942116761351594la_a_b @ P2 )
= ( collec2099942116761351594la_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_72_Collect__cong,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X5: set_nat] :
( ( P2 @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_set_nat @ P2 )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_73_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P2 @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_74_sup_Oidem,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_75_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_76_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_77_sup_Oidem,axiom,
! [A2: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_78_sup_Oidem,axiom,
! [A2: nat > $o] :
( ( sup_sup_nat_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_79_sup_Oidem,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_80_sup__idem,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_81_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_82_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_83_sup__idem,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_84_sup__idem,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_85_sup__idem,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_86_sup_Oleft__idem,axiom,
! [A2: set_b,B: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_87_sup_Oleft__idem,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_88_sup_Oleft__idem,axiom,
! [A2: nat,B: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) )
= ( sup_sup_nat @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_89_sup_Oleft__idem,axiom,
! [A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ A2 @ ( sup_su1471977682094119364_a_b_o @ A2 @ B ) )
= ( sup_su1471977682094119364_a_b_o @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_90_sup_Oleft__idem,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( sup_sup_nat_o @ A2 @ ( sup_sup_nat_o @ A2 @ B ) )
= ( sup_sup_nat_o @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_91_sup_Oleft__idem,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_92_sup__left__idem,axiom,
! [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
= ( sup_sup_set_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_93_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_94_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_95_sup__left__idem,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ X @ Y ) )
= ( sup_su1471977682094119364_a_b_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_96_sup__left__idem,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ X @ Y ) )
= ( sup_sup_nat_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_97_sup__left__idem,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_98_sup_Oright__idem,axiom,
! [A2: set_b,B: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B ) @ B )
= ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_99_sup_Oright__idem,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ B )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_100_sup_Oright__idem,axiom,
! [A2: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B ) @ B )
= ( sup_sup_nat @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_101_sup_Oright__idem,axiom,
! [A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ ( sup_su1471977682094119364_a_b_o @ A2 @ B ) @ B )
= ( sup_su1471977682094119364_a_b_o @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_102_sup_Oright__idem,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( sup_sup_nat_o @ ( sup_sup_nat_o @ A2 @ B ) @ B )
= ( sup_sup_nat_o @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_103_sup_Oright__idem,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ B )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_104_sup__apply,axiom,
( sup_su1471977682094119364_a_b_o
= ( ^ [F2: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o,X3: relational_fmla_a_b] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_apply
thf(fact_105_sup__apply,axiom,
( sup_sup_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o,X3: nat] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_apply
thf(fact_106_finite__UnI,axiom,
! [F: set_list_a,G: set_list_a] :
( ( finite_finite_list_a @ F )
=> ( ( finite_finite_list_a @ G )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_107_finite__UnI,axiom,
! [F: set_se6865892389300016395la_a_b,G: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ F )
=> ( ( finite5238674622262875500la_a_b @ G )
=> ( finite5238674622262875500la_a_b @ ( sup_su4783144482993978935la_a_b @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_108_finite__UnI,axiom,
! [F: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ F )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_109_finite__UnI,axiom,
! [F: set_b,G: set_b] :
( ( finite_finite_b @ F )
=> ( ( finite_finite_b @ G )
=> ( finite_finite_b @ ( sup_sup_set_b @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_110_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_111_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_112_finite__UnI,axiom,
! [F: set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F )
=> ( ( finite5600759454172676150la_a_b @ G )
=> ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_113_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_114_Un__infinite,axiom,
! [S: set_se6865892389300016395la_a_b,T: set_se6865892389300016395la_a_b] :
( ~ ( finite5238674622262875500la_a_b @ S )
=> ~ ( finite5238674622262875500la_a_b @ ( sup_su4783144482993978935la_a_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_115_Un__infinite,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_116_Un__infinite,axiom,
! [S: set_b,T: set_b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_117_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_118_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_119_Un__infinite,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_120_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_121_fv_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(6)
thf(fact_122_sup__fun__def,axiom,
( sup_su1471977682094119364_a_b_o
= ( ^ [F2: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o,X3: relational_fmla_a_b] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_fun_def
thf(fact_123_sup__fun__def,axiom,
( sup_sup_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o,X3: nat] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_fun_def
thf(fact_124_sup__left__commute,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_125_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_126_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_127_sup__left__commute,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ Y @ Z ) )
= ( sup_su1471977682094119364_a_b_o @ Y @ ( sup_su1471977682094119364_a_b_o @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_128_sup__left__commute,axiom,
! [X: nat > $o,Y: nat > $o,Z: nat > $o] :
( ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ Y @ Z ) )
= ( sup_sup_nat_o @ Y @ ( sup_sup_nat_o @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_129_sup__left__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_130_sup_Oleft__commute,axiom,
! [B: set_b,A2: set_b,C: set_b] :
( ( sup_sup_set_b @ B @ ( sup_sup_set_b @ A2 @ C ) )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_131_sup_Oleft__commute,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_132_sup_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A2 @ C ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_133_sup_Oleft__commute,axiom,
! [B: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ B @ ( sup_su1471977682094119364_a_b_o @ A2 @ C ) )
= ( sup_su1471977682094119364_a_b_o @ A2 @ ( sup_su1471977682094119364_a_b_o @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_134_sup_Oleft__commute,axiom,
! [B: nat > $o,A2: nat > $o,C: nat > $o] :
( ( sup_sup_nat_o @ B @ ( sup_sup_nat_o @ A2 @ C ) )
= ( sup_sup_nat_o @ A2 @ ( sup_sup_nat_o @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_135_sup_Oleft__commute,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ C ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_136_sup__commute,axiom,
( sup_sup_set_b
= ( ^ [X3: set_b,Y3: set_b] : ( sup_sup_set_b @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_137_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_138_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_139_sup__commute,axiom,
( sup_su1471977682094119364_a_b_o
= ( ^ [X3: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] : ( sup_su1471977682094119364_a_b_o @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_140_sup__commute,axiom,
( sup_sup_nat_o
= ( ^ [X3: nat > $o,Y3: nat > $o] : ( sup_sup_nat_o @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_141_sup__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_142_sup_Ocommute,axiom,
( sup_sup_set_b
= ( ^ [A3: set_b,B3: set_b] : ( sup_sup_set_b @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_143_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_144_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_145_sup_Ocommute,axiom,
( sup_su1471977682094119364_a_b_o
= ( ^ [A3: relational_fmla_a_b > $o,B3: relational_fmla_a_b > $o] : ( sup_su1471977682094119364_a_b_o @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_146_sup_Ocommute,axiom,
( sup_sup_nat_o
= ( ^ [A3: nat > $o,B3: nat > $o] : ( sup_sup_nat_o @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_147_sup_Ocommute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B3: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_148_sup__assoc,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_149_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_150_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_151_sup__assoc,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ ( sup_su1471977682094119364_a_b_o @ X @ Y ) @ Z )
= ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_152_sup__assoc,axiom,
! [X: nat > $o,Y: nat > $o,Z: nat > $o] :
( ( sup_sup_nat_o @ ( sup_sup_nat_o @ X @ Y ) @ Z )
= ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_153_sup__assoc,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_154_sup_Oassoc,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B ) @ C )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) ) ).
% sup.assoc
thf(fact_155_sup_Oassoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_156_sup_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B ) @ C )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_157_sup_Oassoc,axiom,
! [A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ ( sup_su1471977682094119364_a_b_o @ A2 @ B ) @ C )
= ( sup_su1471977682094119364_a_b_o @ A2 @ ( sup_su1471977682094119364_a_b_o @ B @ C ) ) ) ).
% sup.assoc
thf(fact_158_sup_Oassoc,axiom,
! [A2: nat > $o,B: nat > $o,C: nat > $o] :
( ( sup_sup_nat_o @ ( sup_sup_nat_o @ A2 @ B ) @ C )
= ( sup_sup_nat_o @ A2 @ ( sup_sup_nat_o @ B @ C ) ) ) ).
% sup.assoc
thf(fact_159_sup_Oassoc,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% sup.assoc
thf(fact_160_inf__sup__aci_I5_J,axiom,
( sup_sup_set_b
= ( ^ [X3: set_b,Y3: set_b] : ( sup_sup_set_b @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_161_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_162_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_163_inf__sup__aci_I5_J,axiom,
( sup_su1471977682094119364_a_b_o
= ( ^ [X3: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] : ( sup_su1471977682094119364_a_b_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_164_inf__sup__aci_I5_J,axiom,
( sup_sup_nat_o
= ( ^ [X3: nat > $o,Y3: nat > $o] : ( sup_sup_nat_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_165_inf__sup__aci_I5_J,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_166_inf__sup__aci_I6_J,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_167_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_168_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_169_inf__sup__aci_I6_J,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ ( sup_su1471977682094119364_a_b_o @ X @ Y ) @ Z )
= ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_170_inf__sup__aci_I6_J,axiom,
! [X: nat > $o,Y: nat > $o,Z: nat > $o] :
( ( sup_sup_nat_o @ ( sup_sup_nat_o @ X @ Y ) @ Z )
= ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_171_inf__sup__aci_I6_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_172_inf__sup__aci_I7_J,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_173_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_174_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_175_inf__sup__aci_I7_J,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ Y @ Z ) )
= ( sup_su1471977682094119364_a_b_o @ Y @ ( sup_su1471977682094119364_a_b_o @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_176_inf__sup__aci_I7_J,axiom,
! [X: nat > $o,Y: nat > $o,Z: nat > $o] :
( ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ Y @ Z ) )
= ( sup_sup_nat_o @ Y @ ( sup_sup_nat_o @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_177_inf__sup__aci_I7_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_178_inf__sup__aci_I8_J,axiom,
! [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
= ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_179_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_180_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_181_inf__sup__aci_I8_J,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ ( sup_su1471977682094119364_a_b_o @ X @ Y ) )
= ( sup_su1471977682094119364_a_b_o @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_182_inf__sup__aci_I8_J,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( sup_sup_nat_o @ X @ ( sup_sup_nat_o @ X @ Y ) )
= ( sup_sup_nat_o @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_183_inf__sup__aci_I8_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_184_Un__left__commute,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B2 @ C2 ) )
= ( sup_sup_set_b @ B2 @ ( sup_sup_set_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_185_Un__left__commute,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_186_Un__left__commute,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) )
= ( sup_su5130108678486352897la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_187_Un__left__absorb,axiom,
! [A: set_b,B2: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ A @ B2 ) )
= ( sup_sup_set_b @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_188_Un__left__absorb,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_189_Un__left__absorb,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_190_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B4: set_b] : ( sup_sup_set_b @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_191_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_192_Un__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_193_Un__absorb,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_194_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_195_Un__absorb,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_196_Un__assoc,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A @ B2 ) @ C2 )
= ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_197_Un__assoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_198_Un__assoc,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 )
= ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_199_ball__Un,axiom,
! [A: set_b,B2: set_b,P2: b > $o] :
( ( ! [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_200_ball__Un,axiom,
! [A: set_a,B2: set_a,P2: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_201_ball__Un,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,P2: relational_fmla_a_b > $o] :
( ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_202_bex__Un,axiom,
! [A: set_b,B2: set_b,P2: b > $o] :
( ( ? [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: b] :
( ( member_b @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_203_bex__Un,axiom,
! [A: set_a,B2: set_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_204_bex__Un,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,P2: relational_fmla_a_b > $o] :
( ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_205_UnI2,axiom,
! [C: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_206_UnI2,axiom,
! [C: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ B2 )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ).
% UnI2
thf(fact_207_UnI2,axiom,
! [C: b,B2: set_b,A: set_b] :
( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% UnI2
thf(fact_208_UnI2,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_209_UnI2,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_210_UnI2,axiom,
! [C: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ B2 )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% UnI2
thf(fact_211_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_212_UnI1,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ).
% UnI1
thf(fact_213_UnI1,axiom,
! [C: b,A: set_b,B2: set_b] :
( ( member_b @ C @ A )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% UnI1
thf(fact_214_UnI1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_215_UnI1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_216_UnI1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% UnI1
thf(fact_217_UnE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_218_UnE,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
=> ( ~ ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_219_UnE,axiom,
! [C: b,A: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B2 ) )
=> ( ~ ( member_b @ C @ A )
=> ( member_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_220_UnE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_221_UnE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_222_UnE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
=> ( ~ ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_223_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_224_infinite__Un,axiom,
! [S: set_se6865892389300016395la_a_b,T: set_se6865892389300016395la_a_b] :
( ( ~ ( finite5238674622262875500la_a_b @ ( sup_su4783144482993978935la_a_b @ S @ T ) ) )
= ( ~ ( finite5238674622262875500la_a_b @ S )
| ~ ( finite5238674622262875500la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_225_infinite__Un,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S )
| ~ ( finite1152437895449049373et_nat @ T ) ) ) ).
% infinite_Un
thf(fact_226_infinite__Un,axiom,
! [S: set_b,T: set_b] :
( ( ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) )
= ( ~ ( finite_finite_b @ S )
| ~ ( finite_finite_b @ T ) ) ) ).
% infinite_Un
thf(fact_227_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_228_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_229_infinite__Un,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) ) )
= ( ~ ( finite5600759454172676150la_a_b @ S )
| ~ ( finite5600759454172676150la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_230_cpropagated__simps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_simps(7)
thf(fact_231_sub_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) ).
% sub.simps(6)
thf(fact_232_fv_Osimps_I1_J,axiom,
! [Uu: b,Ts: list_R6823256787227418703term_a] :
( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu @ Ts ) )
= ( relati4569515538964159125_set_a @ Ts ) ) ).
% fv.simps(1)
thf(fact_233_insert__absorb2,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ X @ A ) )
= ( insert7010464514620295119la_a_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_234_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_235_insert__absorb2,axiom,
! [X: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A ) )
= ( insert_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_236_insert__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ X @ A ) )
= ( insert2023870700798818565la_a_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_237_insert__iff,axiom,
! [A2: b,B: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B @ A ) )
= ( ( A2 = B )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_238_insert__iff,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_239_insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B @ A ) )
= ( ( A2 = B )
| ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_240_insert__iff,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
= ( ( A2 = B )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_241_insert__iff,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ A ) )
= ( ( A2 = B )
| ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_242_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_243_insertCI,axiom,
! [A2: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_244_insertCI,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_245_insertCI,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_246_insertCI,axiom,
! [A2: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_247_insertCI,axiom,
! [A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_248_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_249_fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
= ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fmla.inject(7)
thf(fact_250_finite__insert,axiom,
! [A2: b,A: set_b] :
( ( finite_finite_b @ ( insert_b @ A2 @ A ) )
= ( finite_finite_b @ A ) ) ).
% finite_insert
thf(fact_251_finite__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) )
= ( finite_finite_list_a @ A ) ) ).
% finite_insert
thf(fact_252_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_253_finite__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ A ) )
= ( finite5238674622262875500la_a_b @ A ) ) ).
% finite_insert
thf(fact_254_finite__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( finite1152437895449049373et_nat @ A ) ) ).
% finite_insert
thf(fact_255_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_256_finite__insert,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A ) )
= ( finite5600759454172676150la_a_b @ A ) ) ).
% finite_insert
thf(fact_257_Un__insert__left,axiom,
! [A2: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_258_Un__insert__left,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ B2 ) @ C2 )
= ( insert2023870700798818565la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_259_Un__insert__left,axiom,
! [A2: b,B2: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( insert_b @ A2 @ B2 ) @ C2 )
= ( insert_b @ A2 @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_260_Un__insert__left,axiom,
! [A2: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B2 ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_261_Un__insert__left,axiom,
! [A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B2 ) @ C2 )
= ( insert7010464514620295119la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_262_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_263_Un__insert__right,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ ( insert2023870700798818565la_a_b @ A2 @ B2 ) )
= ( insert2023870700798818565la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_264_Un__insert__right,axiom,
! [A: set_b,A2: b,B2: set_b] :
( ( sup_sup_set_b @ A @ ( insert_b @ A2 @ B2 ) )
= ( insert_b @ A2 @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_265_Un__insert__right,axiom,
! [A: set_a,A2: a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B2 ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_266_Un__insert__right,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B2 ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_267_finite__fv__terms__set,axiom,
! [Ts: list_R6823256787227418703term_a] : ( finite_finite_nat @ ( relati4569515538964159125_set_a @ Ts ) ) ).
% finite_fv_terms_set
thf(fact_268_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B5: set_b] :
( ( A
= ( insert_b @ A2 @ B5 ) )
& ~ ( member_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_269_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B5 ) )
& ~ ( member_set_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_270_mk__disjoint__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ? [B5: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ A2 @ B5 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_271_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_272_mk__disjoint__insert,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [B5: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ A2 @ B5 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_273_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_274_insert__commute,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ Y @ A ) )
= ( insert7010464514620295119la_a_b @ Y @ ( insert7010464514620295119la_a_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_275_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_276_insert__commute,axiom,
! [X: b,Y: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A ) )
= ( insert_b @ Y @ ( insert_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_277_insert__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ Y @ A ) )
= ( insert2023870700798818565la_a_b @ Y @ ( insert2023870700798818565la_a_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_278_insert__eq__iff,axiom,
! [A2: b,A: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A2 @ A )
= ( insert_b @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_b] :
( ( A
= ( insert_b @ B @ C3 ) )
& ~ ( member_b @ B @ C3 )
& ( B2
= ( insert_b @ A2 @ C3 ) )
& ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_279_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_280_insert__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ~ ( member3481406638322139244la_a_b @ B @ B2 )
=> ( ( ( insert2023870700798818565la_a_b @ A2 @ A )
= ( insert2023870700798818565la_a_b @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ B @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ B @ C3 )
& ( B2
= ( insert2023870700798818565la_a_b @ A2 @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_281_insert__eq__iff,axiom,
! [A2: a,A: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_282_insert__eq__iff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ B @ B2 )
=> ( ( ( insert7010464514620295119la_a_b @ A2 @ A )
= ( insert7010464514620295119la_a_b @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ B @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ B @ C3 )
& ( B2
= ( insert7010464514620295119la_a_b @ A2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_283_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_284_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_285_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_286_insert__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( insert2023870700798818565la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_287_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_288_insert__absorb,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_289_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_290_insert__ident,axiom,
! [X: b,A: set_b,B2: set_b] :
( ~ ( member_b @ X @ A )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A )
= ( insert_b @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_291_insert__ident,axiom,
! [X: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X @ A )
=> ( ~ ( member_set_nat @ X @ B2 )
=> ( ( ( insert_set_nat @ X @ A )
= ( insert_set_nat @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_292_insert__ident,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ~ ( member3481406638322139244la_a_b @ X @ B2 )
=> ( ( ( insert2023870700798818565la_a_b @ X @ A )
= ( insert2023870700798818565la_a_b @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_293_insert__ident,axiom,
! [X: a,A: set_a,B2: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_294_insert__ident,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A )
= ( insert7010464514620295119la_a_b @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_295_insert__ident,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_296_Set_Oset__insert,axiom,
! [X: b,A: set_b] :
( ( member_b @ X @ A )
=> ~ ! [B5: set_b] :
( ( A
= ( insert_b @ X @ B5 ) )
=> ( member_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_297_Set_Oset__insert,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ! [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ X @ B5 ) )
=> ( member_set_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_298_Set_Oset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
=> ~ ! [B5: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ X @ B5 ) )
=> ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_299_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_300_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ~ ! [B5: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ X @ B5 ) )
=> ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_301_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_302_insertI2,axiom,
! [A2: b,B2: set_b,B: b] :
( ( member_b @ A2 @ B2 )
=> ( member_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_303_insertI2,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_304_insertI2,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ B2 )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_305_insertI2,axiom,
! [A2: a,B2: set_a,B: a] :
( ( member_a @ A2 @ B2 )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_306_insertI2,axiom,
! [A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ B2 )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_307_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_308_insertI1,axiom,
! [A2: b,B2: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B2 ) ) ).
% insertI1
thf(fact_309_insertI1,axiom,
! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_310_insertI1,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ B2 ) ) ).
% insertI1
thf(fact_311_insertI1,axiom,
! [A2: a,B2: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_312_insertI1,axiom,
! [A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ B2 ) ) ).
% insertI1
thf(fact_313_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_314_insertE,axiom,
! [A2: b,B: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_315_insertE,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_316_insertE,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_317_insertE,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_318_insertE,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_319_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_320_sub_Osimps_I7_J,axiom,
! [Z: nat,Q: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).
% sub.simps(7)
thf(fact_321_fmla_Odistinct_I41_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Disj_a_b @ X61 @ X62 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(41)
thf(fact_322_fmla_Odistinct_I35_J,axiom,
! [X4: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(35)
thf(fact_323_fmla_Odistinct_I21_J,axiom,
! [X2: $o,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(21)
thf(fact_324_finite_OinsertI,axiom,
! [A: set_b,A2: b] :
( ( finite_finite_b @ A )
=> ( finite_finite_b @ ( insert_b @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_325_finite_OinsertI,axiom,
! [A: set_list_a,A2: list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_326_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_327_finite_OinsertI,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( finite5238674622262875500la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_328_finite_OinsertI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_329_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_330_finite_OinsertI,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_331_fmla_Odistinct_I11_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(11)
thf(fact_332_equiv__Exists__cong,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
( ( relational_equiv_a_b @ Q @ Q2 )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ ( relati591517084277583526ts_a_b @ X @ Q2 ) ) ) ).
% equiv_Exists_cong
thf(fact_333_sr__False,axiom,
relational_sr_a_b @ ( relational_Bool_a_b @ $false ) ).
% sr_False
thf(fact_334_sr__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sr_a_b @ Q )
=> ( relational_sr_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% sr_cp
thf(fact_335_nocp_Osimps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% nocp.simps(7)
thf(fact_336_equiv__Exists__Disj,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ Q1 ) @ ( relati591517084277583526ts_a_b @ X @ Q22 ) ) ) ).
% equiv_Exists_Disj
thf(fact_337_Exists__nonfree__equiv,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ).
% Exists_nonfree_equiv
thf(fact_338_sub_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).
% sub.simps(4)
thf(fact_339_sr__Disj,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ Q1 )
= ( relational_fv_a_b @ Q22 ) )
=> ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_sr_a_b @ Q1 )
& ( relational_sr_a_b @ Q22 ) ) ) ) ).
% sr_Disj
thf(fact_340_flat__Disj_Osimps_I2_J,axiom,
! [V: b,Va: list_R6823256787227418703term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Pred_b_a @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(2)
thf(fact_341_flat__Disj_Osimps_I7_J,axiom,
! [V: nat,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(7)
thf(fact_342_flat__Disj_Osimps_I3_J,axiom,
! [V: $o] :
( ( restri569617705344514291sj_a_b @ ( relational_Bool_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(3)
thf(fact_343_flat__Disj_Osimps_I5_J,axiom,
! [V: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Neg_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(5)
thf(fact_344_sub_Osimps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] :
( ( relational_sub_a_b @ ( relational_Pred_b_a @ P @ Ts ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(2)
thf(fact_345_eqs__union,axiom,
! [X: nat,X6: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] :
( ( relational_eqs_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X6 @ Y5 ) )
= ( sup_sup_set_nat @ ( relational_eqs_a_b @ X @ X6 ) @ ( relational_eqs_a_b @ X @ Y5 ) ) ) ).
% eqs_union
thf(fact_346_sub_Osimps_I1_J,axiom,
! [T2: $o] :
( ( relational_sub_a_b @ ( relational_Bool_a_b @ T2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(1)
thf(fact_347_sub_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) ).
% sub.simps(5)
thf(fact_348_erase_Osimps_I2_J,axiom,
! [X: nat,Ts: list_R6823256787227418703term_a,P: b] :
( ( ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P @ Ts ) @ X )
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P @ Ts ) @ X )
= ( relational_Pred_b_a @ P @ Ts ) ) ) ) ).
% erase.simps(2)
thf(fact_349_boolean__algebra__cancel_Osup1,axiom,
! [A: set_b,K: set_b,A2: set_b,B: set_b] :
( ( A
= ( sup_sup_set_b @ K @ A2 ) )
=> ( ( sup_sup_set_b @ A @ B )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_350_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_351_boolean__algebra__cancel_Osup1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( sup_sup_nat @ K @ A2 ) )
=> ( ( sup_sup_nat @ A @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_352_boolean__algebra__cancel_Osup1,axiom,
! [A: relational_fmla_a_b > $o,K: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( A
= ( sup_su1471977682094119364_a_b_o @ K @ A2 ) )
=> ( ( sup_su1471977682094119364_a_b_o @ A @ B )
= ( sup_su1471977682094119364_a_b_o @ K @ ( sup_su1471977682094119364_a_b_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_353_boolean__algebra__cancel_Osup1,axiom,
! [A: nat > $o,K: nat > $o,A2: nat > $o,B: nat > $o] :
( ( A
= ( sup_sup_nat_o @ K @ A2 ) )
=> ( ( sup_sup_nat_o @ A @ B )
= ( sup_sup_nat_o @ K @ ( sup_sup_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_354_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A
= ( sup_su5130108678486352897la_a_b @ K @ A2 ) )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_355_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_356_empty__iff,axiom,
! [C: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ C @ bot_bo2891247006866115487la_a_b ) ).
% empty_iff
thf(fact_357_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_358_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_359_empty__iff,axiom,
! [C: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_360_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_361_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_362_all__not__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X3 @ A ) )
= ( A = bot_bo2891247006866115487la_a_b ) ) ).
% all_not_in_conv
thf(fact_363_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_364_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_365_all__not__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_366_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_367_Collect__empty__eq,axiom,
! [P2: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_368_Collect__empty__eq,axiom,
! [P2: set_Re381260168593705685la_a_b > $o] :
( ( ( collec2099942116761351594la_a_b @ P2 )
= bot_bo2891247006866115487la_a_b )
= ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_369_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_370_Collect__empty__eq,axiom,
! [P2: b > $o] :
( ( ( collect_b @ P2 )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_371_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_372_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_373_empty__Collect__eq,axiom,
! [P2: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P2 ) )
= ( ! [X3: relational_fmla_a_b] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_374_empty__Collect__eq,axiom,
! [P2: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ P2 ) )
= ( ! [X3: set_Re381260168593705685la_a_b] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_375_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_376_empty__Collect__eq,axiom,
! [P2: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P2 ) )
= ( ! [X3: b] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_377_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_378_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_379_fmla_Oinject_I5_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
( ( ( relational_Conj_a_b @ X51 @ X52 )
= ( relational_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fmla.inject(5)
thf(fact_380_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_381_singletonI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) ).
% singletonI
thf(fact_382_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_383_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_384_singletonI,axiom,
! [A2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_385_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_386_sup__bot_Oright__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_387_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_388_sup__bot_Oright__neutral,axiom,
! [A2: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ A2 @ bot_bo8852203127187332700_a_b_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_389_sup__bot_Oright__neutral,axiom,
! [A2: nat > $o] :
( ( sup_sup_nat_o @ A2 @ bot_bot_nat_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_390_sup__bot_Oright__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_391_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_b,B: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A2 @ B ) )
= ( ( A2 = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_392_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_393_sup__bot_Oneutr__eq__iff,axiom,
! [A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( bot_bo8852203127187332700_a_b_o
= ( sup_su1471977682094119364_a_b_o @ A2 @ B ) )
= ( ( A2 = bot_bo8852203127187332700_a_b_o )
& ( B = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_394_sup__bot_Oneutr__eq__iff,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ A2 @ B ) )
= ( ( A2 = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_395_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_396_sup__bot_Oleft__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_397_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_398_sup__bot_Oleft__neutral,axiom,
! [A2: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_399_sup__bot_Oleft__neutral,axiom,
! [A2: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_400_sup__bot_Oleft__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_401_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_402_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_403_sup__bot_Oeq__neutr__iff,axiom,
! [A2: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( ( sup_su1471977682094119364_a_b_o @ A2 @ B )
= bot_bo8852203127187332700_a_b_o )
= ( ( A2 = bot_bo8852203127187332700_a_b_o )
& ( B = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_404_sup__bot_Oeq__neutr__iff,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( ( sup_sup_nat_o @ A2 @ B )
= bot_bot_nat_o )
= ( ( A2 = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_405_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_406_sup__eq__bot__iff,axiom,
! [X: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X @ Y )
= bot_bot_set_b )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_407_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_408_sup__eq__bot__iff,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( ( sup_su1471977682094119364_a_b_o @ X @ Y )
= bot_bo8852203127187332700_a_b_o )
= ( ( X = bot_bo8852203127187332700_a_b_o )
& ( Y = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_409_sup__eq__bot__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ( sup_sup_nat_o @ X @ Y )
= bot_bot_nat_o )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_410_sup__eq__bot__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_411_bot__eq__sup__iff,axiom,
! [X: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X @ Y ) )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_412_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_413_bot__eq__sup__iff,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( bot_bo8852203127187332700_a_b_o
= ( sup_su1471977682094119364_a_b_o @ X @ Y ) )
= ( ( X = bot_bo8852203127187332700_a_b_o )
& ( Y = bot_bo8852203127187332700_a_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_414_bot__eq__sup__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ X @ Y ) )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_415_bot__eq__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_416_sup__bot__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% sup_bot_right
thf(fact_417_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_418_sup__bot__right,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ bot_bo8852203127187332700_a_b_o )
= X ) ).
% sup_bot_right
thf(fact_419_sup__bot__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% sup_bot_right
thf(fact_420_sup__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% sup_bot_right
thf(fact_421_sup__bot__left,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X )
= X ) ).
% sup_bot_left
thf(fact_422_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_423_sup__bot__left,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o @ X )
= X ) ).
% sup_bot_left
thf(fact_424_sup__bot__left,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ X )
= X ) ).
% sup_bot_left
thf(fact_425_sup__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_426_Un__empty,axiom,
! [A: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A @ B2 )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_427_Un__empty,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_428_Un__empty,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= bot_bo4495933725496725865la_a_b )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Un_empty
thf(fact_429_cpropagated__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q22 ) ) ) ).
% cpropagated_simps(5)
thf(fact_430_erase_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X )
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q22 @ X ) ) ) ).
% erase.simps(5)
thf(fact_431_eqs__noteq,axiom,
! [Y: nat,X: nat,Q: set_Re381260168593705685la_a_b] :
( ( member_nat @ Y @ ( relational_eqs_a_b @ X @ Q ) )
=> ( X != Y ) ) ).
% eqs_noteq
thf(fact_432_not__self__eqs,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
~ ( member_nat @ X @ ( relational_eqs_a_b @ X @ G ) ) ).
% not_self_eqs
thf(fact_433_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_434_emptyE,axiom,
! [A2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ).
% emptyE
thf(fact_435_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_436_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_437_emptyE,axiom,
! [A2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_438_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_439_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_440_equals0D,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( A = bot_bo2891247006866115487la_a_b )
=> ~ ( member3481406638322139244la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_441_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_442_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_443_equals0D,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( A = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_444_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_445_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y6: set_nat] :
~ ( member_set_nat @ Y6 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_446_equals0I,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ! [Y6: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ Y6 @ A )
=> ( A = bot_bo2891247006866115487la_a_b ) ) ).
% equals0I
thf(fact_447_equals0I,axiom,
! [A: set_b] :
( ! [Y6: b] :
~ ( member_b @ Y6 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_448_equals0I,axiom,
! [A: set_a] :
( ! [Y6: a] :
~ ( member_a @ Y6 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_449_equals0I,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [Y6: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y6 @ A )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_450_equals0I,axiom,
! [A: set_nat] :
( ! [Y6: nat] :
~ ( member_nat @ Y6 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_451_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_452_ex__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ? [X3: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X3 @ A ) )
= ( A != bot_bo2891247006866115487la_a_b ) ) ).
% ex_in_conv
thf(fact_453_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_454_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_455_ex__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ? [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A ) )
= ( A != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_456_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_457_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_458_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_459_boolean__algebra_Odisj__zero__right,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ bot_bo8852203127187332700_a_b_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_460_boolean__algebra_Odisj__zero__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_461_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_462_erase_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X )
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q22 @ X ) ) ) ).
% erase.simps(6)
thf(fact_463_erase_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Neg_a_b @ Q ) @ X )
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% erase.simps(4)
thf(fact_464_erase_Osimps_I1_J,axiom,
! [T2: $o,X: nat] :
( ( relational_erase_a_b @ ( relational_Bool_a_b @ T2 ) @ X )
= ( relational_Bool_a_b @ T2 ) ) ).
% erase.simps(1)
thf(fact_465_erase_Osimps_I7_J,axiom,
! [X: nat,Z: nat,Q: relational_fmla_a_b] :
( ( ( X = Z )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ X @ Q ) ) )
& ( ( X != Z )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q @ X ) ) ) ) ) ).
% erase.simps(7)
thf(fact_466_fmla_Odistinct_I37_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(37)
thf(fact_467_fmla_Odistinct_I31_J,axiom,
! [X4: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(31)
thf(fact_468_fmla_Odistinct_I17_J,axiom,
! [X2: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(17)
thf(fact_469_fmla_Odistinct_I39_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(39)
thf(fact_470_fmla_Odistinct_I7_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(7)
thf(fact_471_flat__Disj_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Conj_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(6)
thf(fact_472_nocp_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q22 ) ) ) ).
% nocp.simps(5)
thf(fact_473_singletonD,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_474_singletonD,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_475_singletonD,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_476_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_477_singletonD,axiom,
! [B: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_478_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_479_singleton__iff,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_480_singleton__iff,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_481_singleton__iff,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_482_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_483_singleton__iff,axiom,
! [B: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_484_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_485_doubleton__eq__iff,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b,C: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_486_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_487_doubleton__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B @ bot_bo2891247006866115487la_a_b ) )
= ( insert2023870700798818565la_a_b @ C @ ( insert2023870700798818565la_a_b @ D @ bot_bo2891247006866115487la_a_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_488_doubleton__eq__iff,axiom,
! [A2: b,B: b,C: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_489_doubleton__eq__iff,axiom,
! [A2: a,B: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_490_insert__not__empty,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ A )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_491_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_492_insert__not__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ A2 @ A )
!= bot_bo2891247006866115487la_a_b ) ).
% insert_not_empty
thf(fact_493_insert__not__empty,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_494_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_495_singleton__inject,axiom,
! [A2: relational_fmla_a_b,B: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_496_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_497_singleton__inject,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= ( insert2023870700798818565la_a_b @ B @ bot_bo2891247006866115487la_a_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_498_singleton__inject,axiom,
! [A2: b,B: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_499_singleton__inject,axiom,
! [A2: a,B: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_500_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_501_infinite__imp__nonempty,axiom,
! [S: set_se6865892389300016395la_a_b] :
( ~ ( finite5238674622262875500la_a_b @ S )
=> ( S != bot_bo2891247006866115487la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_502_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_503_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_504_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_505_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_506_infinite__imp__nonempty,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_507_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_508_finite_OemptyI,axiom,
finite5238674622262875500la_a_b @ bot_bo2891247006866115487la_a_b ).
% finite.emptyI
thf(fact_509_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_510_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_511_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_512_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_513_finite_OemptyI,axiom,
finite5600759454172676150la_a_b @ bot_bo4495933725496725865la_a_b ).
% finite.emptyI
thf(fact_514_equiv__Conj__cong,axiom,
! [Q1: relational_fmla_a_b,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q12 )
=> ( ( relational_equiv_a_b @ Q22 @ Q23 )
=> ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) ) ) ).
% equiv_Conj_cong
thf(fact_515_Un__empty__right,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% Un_empty_right
thf(fact_516_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_517_Un__empty__right,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% Un_empty_right
thf(fact_518_Un__empty__left,axiom,
! [B2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_519_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_520_Un__empty__left,axiom,
! [B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_521_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A5: set_list_a] :
( ? [A6: list_a] :
( A2
= ( insert_list_a @ A6 @ A5 ) )
=> ~ ( finite_finite_list_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_522_finite_Ocases,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ~ ! [A5: set_se6865892389300016395la_a_b] :
( ? [A6: set_Re381260168593705685la_a_b] :
( A2
= ( insert2023870700798818565la_a_b @ A6 @ A5 ) )
=> ~ ( finite5238674622262875500la_a_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_523_finite_Ocases,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ~ ! [A5: set_set_nat] :
( ? [A6: set_nat] :
( A2
= ( insert_set_nat @ A6 @ A5 ) )
=> ~ ( finite1152437895449049373et_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_524_finite_Ocases,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( A2 != bot_bot_set_b )
=> ~ ! [A5: set_b] :
( ? [A6: b] :
( A2
= ( insert_b @ A6 @ A5 ) )
=> ~ ( finite_finite_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_525_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A6: a] :
( A2
= ( insert_a @ A6 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_526_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A6: nat] :
( A2
= ( insert_nat @ A6 @ A5 ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_527_finite_Ocases,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ~ ! [A5: set_Re381260168593705685la_a_b] :
( ? [A6: relational_fmla_a_b] :
( A2
= ( insert7010464514620295119la_a_b @ A6 @ A5 ) )
=> ~ ( finite5600759454172676150la_a_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_528_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A3: set_list_a] :
( ( A3 = bot_bot_set_list_a )
| ? [A4: set_list_a,B3: list_a] :
( ( A3
= ( insert_list_a @ B3 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_529_finite_Osimps,axiom,
( finite5238674622262875500la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( ( A3 = bot_bo2891247006866115487la_a_b )
| ? [A4: set_se6865892389300016395la_a_b,B3: set_Re381260168593705685la_a_b] :
( ( A3
= ( insert2023870700798818565la_a_b @ B3 @ A4 ) )
& ( finite5238674622262875500la_a_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_530_finite_Osimps,axiom,
( finite1152437895449049373et_nat
= ( ^ [A3: set_set_nat] :
( ( A3 = bot_bot_set_set_nat )
| ? [A4: set_set_nat,B3: set_nat] :
( ( A3
= ( insert_set_nat @ B3 @ A4 ) )
& ( finite1152437895449049373et_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_531_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A3: set_b] :
( ( A3 = bot_bot_set_b )
| ? [A4: set_b,B3: b] :
( ( A3
= ( insert_b @ B3 @ A4 ) )
& ( finite_finite_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_532_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A4: set_a,B3: a] :
( ( A3
= ( insert_a @ B3 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_533_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A4: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_534_finite_Osimps,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( ( A3 = bot_bo4495933725496725865la_a_b )
| ? [A4: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b] :
( ( A3
= ( insert7010464514620295119la_a_b @ B3 @ A4 ) )
& ( finite5600759454172676150la_a_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_535_finite__induct,axiom,
! [F: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_536_finite__induct,axiom,
! [F: set_se6865892389300016395la_a_b,P2: set_se6865892389300016395la_a_b > $o] :
( ( finite5238674622262875500la_a_b @ F )
=> ( ( P2 @ bot_bo2891247006866115487la_a_b )
=> ( ! [X5: set_Re381260168593705685la_a_b,F3: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ F3 )
=> ( ~ ( member3481406638322139244la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert2023870700798818565la_a_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_537_finite__induct,axiom,
! [F: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X5: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_nat @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_538_finite__induct,axiom,
! [F: set_b,P2: set_b > $o] :
( ( finite_finite_b @ F )
=> ( ( P2 @ bot_bot_set_b )
=> ( ! [X5: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_539_finite__induct,axiom,
! [F: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_540_finite__induct,axiom,
! [F: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_541_finite__induct,axiom,
! [F: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ~ ( member4680049679412964150la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_542_finite__ne__induct,axiom,
! [F: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( F != bot_bot_set_list_a )
=> ( ! [X5: list_a] : ( P2 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
=> ( ! [X5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_543_finite__ne__induct,axiom,
! [F: set_se6865892389300016395la_a_b,P2: set_se6865892389300016395la_a_b > $o] :
( ( finite5238674622262875500la_a_b @ F )
=> ( ( F != bot_bo2891247006866115487la_a_b )
=> ( ! [X5: set_Re381260168593705685la_a_b] : ( P2 @ ( insert2023870700798818565la_a_b @ X5 @ bot_bo2891247006866115487la_a_b ) )
=> ( ! [X5: set_Re381260168593705685la_a_b,F3: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ F3 )
=> ( ( F3 != bot_bo2891247006866115487la_a_b )
=> ( ~ ( member3481406638322139244la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert2023870700798818565la_a_b @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_544_finite__ne__induct,axiom,
! [F: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F )
=> ( ( F != bot_bot_set_set_nat )
=> ( ! [X5: set_nat] : ( P2 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
=> ( ! [X5: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( F3 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_nat @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_545_finite__ne__induct,axiom,
! [F: set_b,P2: set_b > $o] :
( ( finite_finite_b @ F )
=> ( ( F != bot_bot_set_b )
=> ( ! [X5: b] : ( P2 @ ( insert_b @ X5 @ bot_bot_set_b ) )
=> ( ! [X5: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_b @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_546_finite__ne__induct,axiom,
! [F: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X5: a] : ( P2 @ ( insert_a @ X5 @ bot_bot_set_a ) )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_547_finite__ne__induct,axiom,
! [F: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X5: nat] : ( P2 @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_548_finite__ne__induct,axiom,
! [F: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F )
=> ( ( F != bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b] : ( P2 @ ( insert7010464514620295119la_a_b @ X5 @ bot_bo4495933725496725865la_a_b ) )
=> ( ! [X5: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( F3 != bot_bo4495933725496725865la_a_b )
=> ( ~ ( member4680049679412964150la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ X5 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_549_infinite__finite__induct,axiom,
! [P2: set_list_a > $o,A: set_list_a] :
( ! [A5: set_list_a] :
( ~ ( finite_finite_list_a @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_550_infinite__finite__induct,axiom,
! [P2: set_se6865892389300016395la_a_b > $o,A: set_se6865892389300016395la_a_b] :
( ! [A5: set_se6865892389300016395la_a_b] :
( ~ ( finite5238674622262875500la_a_b @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bo2891247006866115487la_a_b )
=> ( ! [X5: set_Re381260168593705685la_a_b,F3: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ F3 )
=> ( ~ ( member3481406638322139244la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert2023870700798818565la_a_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_551_infinite__finite__induct,axiom,
! [P2: set_set_nat > $o,A: set_set_nat] :
( ! [A5: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X5: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_nat @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_552_infinite__finite__induct,axiom,
! [P2: set_b > $o,A: set_b] :
( ! [A5: set_b] :
( ~ ( finite_finite_b @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bot_set_b )
=> ( ! [X5: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_553_infinite__finite__induct,axiom,
! [P2: set_a > $o,A: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_554_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ! [A5: set_nat] :
( ~ ( finite_finite_nat @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_555_infinite__finite__induct,axiom,
! [P2: set_Re381260168593705685la_a_b > $o,A: set_Re381260168593705685la_a_b] :
( ! [A5: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A5 )
=> ( P2 @ A5 ) )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ~ ( member4680049679412964150la_a_b @ X5 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ X5 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_556_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_557_insert__is__Un,axiom,
( insert2023870700798818565la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] : ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A3 @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_558_insert__is__Un,axiom,
( insert_b
= ( ^ [A3: b] : ( sup_sup_set_b @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_559_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_560_insert__is__Un,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_561_Un__singleton__iff,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_562_Un__singleton__iff,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A @ B2 )
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= ( ( ( A = bot_bo2891247006866115487la_a_b )
& ( B2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B2 = bot_bo2891247006866115487la_a_b ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_563_Un__singleton__iff,axiom,
! [A: set_b,B2: set_b,X: b] :
( ( ( sup_sup_set_b @ A @ B2 )
= ( insert_b @ X @ bot_bot_set_b ) )
= ( ( ( A = bot_bot_set_b )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2 = bot_bot_set_b ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_564_Un__singleton__iff,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_565_Un__singleton__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B2 = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_566_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( ( A = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_567_singleton__Un__iff,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( ( ( A = bot_bo2891247006866115487la_a_b )
& ( B2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B2 = bot_bo2891247006866115487la_a_b ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_568_singleton__Un__iff,axiom,
! [X: b,A: set_b,B2: set_b] :
( ( ( insert_b @ X @ bot_bot_set_b )
= ( sup_sup_set_b @ A @ B2 ) )
= ( ( ( A = bot_bot_set_b )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2 = bot_bot_set_b ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_569_singleton__Un__iff,axiom,
! [X: a,A: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B2 ) )
= ( ( ( A = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_570_singleton__Un__iff,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B2 = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_571_fv_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(5)
thf(fact_572_finite__eqs,axiom,
! [G: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ G )
=> ( finite_finite_nat @ ( relational_eqs_a_b @ X @ G ) ) ) ).
% finite_eqs
thf(fact_573_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_b,K: set_b,B: set_b,A2: set_b] :
( ( B2
= ( sup_sup_set_b @ K @ B ) )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_574_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_a,K: set_a,B: set_a,A2: set_a] :
( ( B2
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_575_boolean__algebra__cancel_Osup2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A2 @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_576_boolean__algebra__cancel_Osup2,axiom,
! [B2: relational_fmla_a_b > $o,K: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o] :
( ( B2
= ( sup_su1471977682094119364_a_b_o @ K @ B ) )
=> ( ( sup_su1471977682094119364_a_b_o @ A2 @ B2 )
= ( sup_su1471977682094119364_a_b_o @ K @ ( sup_su1471977682094119364_a_b_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_577_boolean__algebra__cancel_Osup2,axiom,
! [B2: nat > $o,K: nat > $o,B: nat > $o,A2: nat > $o] :
( ( B2
= ( sup_sup_nat_o @ K @ B ) )
=> ( ( sup_sup_nat_o @ A2 @ B2 )
= ( sup_sup_nat_o @ K @ ( sup_sup_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_578_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( sup_su5130108678486352897la_a_b @ K @ B ) )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B2 )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_579_flat__Disj_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X )
= Y )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q13 ) @ ( restri569617705344514291sj_a_b @ Q24 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.elims
thf(fact_580_sub_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y )
=> ( ! [T3: $o] :
( ( X
= ( relational_Bool_a_b @ T3 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).
% sub.elims
thf(fact_581_the__elem__eq,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_582_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_583_the__elem__eq,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( the_el7486773796875720352la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_584_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_585_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_586_bot__apply,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X3: relational_fmla_a_b] : bot_bot_o ) ) ).
% bot_apply
thf(fact_587_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_588_is__singletonI,axiom,
! [X: set_Re381260168593705685la_a_b] : ( is_sin1114528679824004833la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ).
% is_singletonI
thf(fact_589_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_590_is__singletonI,axiom,
! [X: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_591_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_592_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_593_finite__transitivity__chain,axiom,
! [A: set_list_a,R: list_a > list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ! [X5: list_a] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: list_a,Y6: list_a,Z2: list_a] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A )
=> ? [Y7: list_a] :
( ( member_list_a @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bot_set_list_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_594_finite__transitivity__chain,axiom,
! [A: set_se6865892389300016395la_a_b,R: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ! [X5: set_Re381260168593705685la_a_b] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X5 @ A )
=> ? [Y7: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_595_finite__transitivity__chain,axiom,
! [A: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X5: set_nat] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: set_nat,Y6: set_nat,Z2: set_nat] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ? [Y7: set_nat] :
( ( member_set_nat @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_596_finite__transitivity__chain,axiom,
! [A: set_b,R: b > b > $o] :
( ( finite_finite_b @ A )
=> ( ! [X5: b] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: b,Y6: b,Z2: b] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A )
=> ? [Y7: b] :
( ( member_b @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bot_set_b ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_597_finite__transitivity__chain,axiom,
! [A: set_a,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X5: a] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: a,Y6: a,Z2: a] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A )
=> ? [Y7: a] :
( ( member_a @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_598_finite__transitivity__chain,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X5: nat] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y6: nat,Z2: nat] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Y7: nat] :
( ( member_nat @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_599_finite__transitivity__chain,axiom,
! [A: set_Re381260168593705685la_a_b,R: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ! [X5: relational_fmla_a_b] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: relational_fmla_a_b,Y6: relational_fmla_a_b,Z2: relational_fmla_a_b] :
( ( R @ X5 @ Y6 )
=> ( ( R @ Y6 @ Z2 )
=> ( R @ X5 @ Z2 ) ) )
=> ( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ? [Y7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y7 @ A )
& ( R @ X5 @ Y7 ) ) )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_600_fmla_Oexhaust,axiom,
! [Y: relational_fmla_a_b] :
( ! [X112: b,X122: list_R6823256787227418703term_a] :
( Y
!= ( relational_Pred_b_a @ X112 @ X122 ) )
=> ( ! [X22: $o] :
( Y
!= ( relational_Bool_a_b @ X22 ) )
=> ( ! [X31: nat,X32: relational_term_a] :
( Y
!= ( relational_Eq_a_b @ X31 @ X32 ) )
=> ( ! [X42: relational_fmla_a_b] :
( Y
!= ( relational_Neg_a_b @ X42 ) )
=> ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
( Y
!= ( relational_Conj_a_b @ X512 @ X522 ) )
=> ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
( Y
!= ( relational_Disj_a_b @ X612 @ X622 ) )
=> ~ ! [X712: nat,X722: relational_fmla_a_b] :
( Y
!= ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).
% fmla.exhaust
thf(fact_601_fmla_Oinject_I3_J,axiom,
! [X312: nat,X322: relational_term_a,Y31: nat,Y32: relational_term_a] :
( ( ( relational_Eq_a_b @ X312 @ X322 )
= ( relational_Eq_a_b @ Y31 @ Y32 ) )
= ( ( X312 = Y31 )
& ( X322 = Y32 ) ) ) ).
% fmla.inject(3)
thf(fact_602_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_603_fmla_Odistinct_I27_J,axiom,
! [X312: nat,X322: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X312 @ X322 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(27)
thf(fact_604_fmla_Odistinct_I13_J,axiom,
! [X2: $o,X312: nat,X322: relational_term_a] :
( ( relational_Bool_a_b @ X2 )
!= ( relational_Eq_a_b @ X312 @ X322 ) ) ).
% fmla.distinct(13)
thf(fact_605_fmla_Odistinct_I3_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X312: nat,X322: relational_term_a] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Eq_a_b @ X312 @ X322 ) ) ).
% fmla.distinct(3)
thf(fact_606_fv_Osimps_I2_J,axiom,
! [B: $o] :
( ( relational_fv_a_b @ ( relational_Bool_a_b @ B ) )
= bot_bot_set_nat ) ).
% fv.simps(2)
thf(fact_607_is__singletonI_H,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b,Y6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ( ( member4680049679412964150la_a_b @ Y6 @ A )
=> ( X5 = Y6 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_608_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X5: nat,Y6: nat] :
( ( member_nat @ X5 @ A )
=> ( ( member_nat @ Y6 @ A )
=> ( X5 = Y6 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_609_sub_Osimps_I3_J,axiom,
! [X: nat,T2: relational_term_a] :
( ( relational_sub_a_b @ ( relational_Eq_a_b @ X @ T2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(3)
thf(fact_610_flat__Disj_Osimps_I4_J,axiom,
! [V: nat,Va: relational_term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Eq_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(4)
thf(fact_611_cp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X5: nat,T3: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ! [Q4: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q4 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ~ ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).
% cp.cases
thf(fact_612_fv_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
=> ( ! [B6: $o] :
( X
!= ( relational_Bool_a_b @ B6 ) )
=> ( ! [X5: nat,T4: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X5 @ T4 ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Phi2 ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) ) ) ) ) ) ) ) ).
% fv.cases
thf(fact_613_nocp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [B6: $o] :
( X
!= ( relational_Bool_a_b @ B6 ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ! [Q4: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q4 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) ) ) ) ) ) ) ) ).
% nocp.cases
thf(fact_614_flat__Disj_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( X
!= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% flat_Disj.cases
thf(fact_615_nocp_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relational_nocp_a_b @ X )
= Y )
=> ( ( ? [B6: $o] :
( X
= ( relational_Bool_a_b @ B6 ) )
=> Y )
=> ( ( ? [P3: b,Ts2: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ~ Y )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( Y
= ( T3
= ( relational_Var_a @ X5 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( Y
= ( ~ ( relational_nocp_a_b @ Q4 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( Y
= ( ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
= ( ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( Y
= ( ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.elims(1)
thf(fact_616_nocp_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( T3
= ( relational_Var_a @ X5 ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ~ ( relational_nocp_a_b @ Q4 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(2)
thf(fact_617_nocp_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ! [B6: $o] :
( X
!= ( relational_Bool_a_b @ B6 ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( T3
!= ( relational_Var_a @ X5 ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( relational_nocp_a_b @ Q4 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(3)
thf(fact_618_erase_Ocases,axiom,
! [X: produc7366699395886430672_b_nat] :
( ! [T3: $o,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T3 ) @ X5 ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ X5 ) )
=> ( ! [Z2: nat,T3: relational_term_a,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X5 ) )
=> ( ! [Q4: relational_fmla_a_b,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q4 ) @ X5 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ X5 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ X5 ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b,X5: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) @ X5 ) ) ) ) ) ) ) ) ).
% erase.cases
thf(fact_619_flat__Disj_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ X )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q13 ) @ ( restri569617705344514291sj_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.pelims
thf(fact_620_sub_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X )
=> ( ! [T3: $o] :
( ( X
= ( relational_Bool_a_b @ T3 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T3 ) ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sub.pelims
thf(fact_621_fv_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y )
=> ( ! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
=> ( Y
!= ( relati4569515538964159125_set_a @ Ts2 ) ) )
=> ( ( ? [B6: $o] :
( X
= ( relational_Bool_a_b @ B6 ) )
=> ( Y != bot_bot_set_nat ) )
=> ( ! [X5: nat,T4: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T4 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T4 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi2 ) )
=> ( Y
!= ( relational_fv_a_b @ Phi2 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).
% fv.elims
thf(fact_622_DiffI,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( ~ ( member4680049679412964150la_a_b @ C @ B2 )
=> ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_623_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_624_Diff__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B2 ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
& ~ ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_625_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_626_Diff__insert0,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B2 ) )
= ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_627_Diff__insert0,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_628_insert__Diff1,axiom,
! [X: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B2 )
= ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_629_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_630_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_631_finite__Diff2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B2 )
=> ( ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B2 ) )
= ( finite5600759454172676150la_a_b @ A ) ) ) ).
% finite_Diff2
thf(fact_632_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_633_finite__Diff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_634_Un__Diff__cancel2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ B2 @ A ) @ A )
= ( sup_su5130108678486352897la_a_b @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_635_Un__Diff__cancel,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B2 @ A ) )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_636_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_637_finite__Diff__insert,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B2 ) ) )
= ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_638_cpropagated__simps_I3_J,axiom,
! [X: nat,T2: relational_term_a] :
( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X @ T2 ) )
= ( T2
!= ( relational_Var_a @ X ) ) ) ).
% cpropagated_simps(3)
thf(fact_639_DiffE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B2 ) )
=> ~ ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_640_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_641_DiffD1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B2 ) )
=> ( member4680049679412964150la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_642_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_643_DiffD2,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B2 ) )
=> ~ ( member4680049679412964150la_a_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_644_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_645_insert__Diff__if,axiom,
! [X: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B2 )
= ( minus_4077726661957047470la_a_b @ A @ B2 ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B2 )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_646_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_647_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_648_Diff__infinite__finite,axiom,
! [T: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ T )
=> ( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_649_Un__Diff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 )
= ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A @ C2 ) @ ( minus_4077726661957047470la_a_b @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_650_erase_Osimps_I3_J,axiom,
! [T2: relational_term_a,Z: nat,X: nat] :
( ( ( T2
= ( relational_Var_a @ Z ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z @ T2 ) @ X )
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T2
!= ( relational_Var_a @ Z ) )
=> ( ( ( ( X = Z )
| ( member_nat @ X @ ( relati6004689760767320788_set_a @ T2 ) ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z @ T2 ) @ X )
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( X = Z )
| ( member_nat @ X @ ( relati6004689760767320788_set_a @ T2 ) ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z @ T2 ) @ X )
= ( relational_Eq_a_b @ Z @ T2 ) ) ) ) ) ) ).
% erase.simps(3)
thf(fact_651_Diff__insert__absorb,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_652_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_653_insert__Diff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_654_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_655_nocp_Osimps_I3_J,axiom,
! [X: nat,T2: relational_term_a] :
( ( relational_nocp_a_b @ ( relational_Eq_a_b @ X @ T2 ) )
= ( T2
!= ( relational_Var_a @ X ) ) ) ).
% nocp.simps(3)
thf(fact_656_notin__eqs,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ~ ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) ) ) ).
% notin_eqs
thf(fact_657_eqs__in,axiom,
! [Y: nat,X: nat,G: set_Re381260168593705685la_a_b] :
( ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) )
=> ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G ) ) ).
% eqs_in
thf(fact_658_fv_Osimps_I7_J,axiom,
! [Z: nat,Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ).
% fv.simps(7)
thf(fact_659_finite__empty__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ A )
=> ( ! [A6: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ A6 @ A5 )
=> ( ( P2 @ A5 )
=> ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_660_finite__empty__induct,axiom,
! [A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( P2 @ A )
=> ( ! [A6: relational_fmla_a_b,A5: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A5 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A5 )
=> ( ( P2 @ A5 )
=> ( P2 @ ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ A6 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ( P2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_661_infinite__coinduct,axiom,
! [X6: set_nat > $o,A: set_nat] :
( ( X6 @ A )
=> ( ! [A5: set_nat] :
( ( X6 @ A5 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A5 )
& ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_662_infinite__coinduct,axiom,
! [X6: set_Re381260168593705685la_a_b > $o,A: set_Re381260168593705685la_a_b] :
( ( X6 @ A )
=> ( ! [A5: set_Re381260168593705685la_a_b] :
( ( X6 @ A5 )
=> ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A5 )
& ( ( X6 @ ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) )
| ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ~ ( finite5600759454172676150la_a_b @ A ) ) ) ).
% infinite_coinduct
thf(fact_663_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_664_infinite__remove,axiom,
! [S: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% infinite_remove
thf(fact_665_sr__Conj__eq,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_sr_a_b @ Q )
=> ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
| ( member_nat @ Y @ ( relational_fv_a_b @ Q ) ) )
=> ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ) ).
% sr_Conj_eq
thf(fact_666_fv_Osimps_I3_J,axiom,
! [X: nat,T5: relational_term_a] :
( ( relational_fv_a_b @ ( relational_Eq_a_b @ X @ T5 ) )
= ( sup_sup_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) ) ).
% fv.simps(3)
thf(fact_667_erase_Oelims,axiom,
! [X: relational_fmla_a_b,Xa: nat,Y: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa )
= Y )
=> ( ! [T3: $o] :
( ( X
= ( relational_Bool_a_b @ T3 ) )
=> ( Y
!= ( relational_Bool_a_b @ T3 ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ~ ( ( ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( Y
= ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) ) )
=> ( ! [Z2: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z2 @ T3 ) )
=> ~ ( ( ( T3
= ( relational_Var_a @ Z2 ) )
=> ( Y
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T3
!= ( relational_Var_a @ Z2 ) )
=> ( ( ( ( Xa = Z2 )
| ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa = Z2 )
| ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( Y
= ( relational_Eq_a_b @ Z2 @ T3 ) ) ) ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( Y
!= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q13 @ Xa ) @ ( relational_erase_a_b @ Q24 @ Xa ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q13 @ Xa ) @ ( relational_erase_a_b @ Q24 @ Xa ) ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) )
=> ~ ( ( ( Xa = Z2 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
& ( ( Xa != Z2 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z2 @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.elims
thf(fact_668_fv_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X )
=> ( ! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
=> ( ( Y
= ( relati4569515538964159125_set_a @ Ts2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts2 ) ) ) )
=> ( ! [B6: $o] :
( ( X
= ( relational_Bool_a_b @ B6 ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B6 ) ) ) )
=> ( ! [X5: nat,T4: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T4 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X5 @ T4 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi2 ) )
=> ( ( Y
= ( relational_fv_a_b @ Phi2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv.pelims
thf(fact_669_nocp_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relational_nocp_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B6: $o] :
( ( X
= ( relational_Bool_a_b @ B6 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B6 ) ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ( Y
= ( T3
!= ( relational_Var_a @ X5 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( Y
= ( relational_nocp_a_b @ Q4 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( Y
= ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(1)
thf(fact_670_nocp_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( T3
= ( relational_Var_a @ X5 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
=> ~ ( relational_nocp_a_b @ Q4 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(2)
thf(fact_671_nocp_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B6: $o] :
( ( X
= ( relational_Bool_a_b @ B6 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B6 ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( T3
!= ( relational_Var_a @ X5 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
=> ( relational_nocp_a_b @ Q4 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q13 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(3)
thf(fact_672_set__minus__singleton__eq,axiom,
! [X: relational_fmla_a_b,X6: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ X6 )
=> ( ( minus_4077726661957047470la_a_b @ X6 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X6 ) ) ).
% set_minus_singleton_eq
thf(fact_673_set__minus__singleton__eq,axiom,
! [X: nat,X6: set_nat] :
( ~ ( member_nat @ X @ X6 )
=> ( ( minus_minus_set_nat @ X6 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X6 ) ) ).
% set_minus_singleton_eq
thf(fact_674_set__notEmptyE,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ! [X5: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X5 @ S ) ) ).
% set_notEmptyE
thf(fact_675_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X5: nat] :
~ ( member_nat @ X5 @ S ) ) ).
% set_notEmptyE
thf(fact_676_memb__imp__not__empty,axiom,
! [X: relational_fmla_a_b,S: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% memb_imp_not_empty
thf(fact_677_memb__imp__not__empty,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_678_set__diff__diff__left,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B2 ) @ C2 )
= ( minus_4077726661957047470la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% set_diff_diff_left
thf(fact_679_insert__remove__id,axiom,
! [X: relational_fmla_a_b,X6: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ X6 )
=> ( X6
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ X6 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_680_insert__remove__id,axiom,
! [X: nat,X6: set_nat] :
( ( member_nat @ X @ X6 )
=> ( X6
= ( insert_nat @ X @ ( minus_minus_set_nat @ X6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_681_erase_Opelims,axiom,
! [X: relational_fmla_a_b,Xa: nat,Y: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa )
= Y )
=> ( ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ X @ Xa ) )
=> ( ! [T3: $o] :
( ( X
= ( relational_Bool_a_b @ T3 ) )
=> ( ( Y
= ( relational_Bool_a_b @ T3 ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T3 ) @ Xa ) ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( ( ( ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( Y
= ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ Xa ) ) ) )
=> ( ! [Z2: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z2 @ T3 ) )
=> ( ( ( ( T3
= ( relational_Var_a @ Z2 ) )
=> ( Y
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T3
!= ( relational_Var_a @ Z2 ) )
=> ( ( ( ( Xa = Z2 )
| ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa = Z2 )
| ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( Y
= ( relational_Eq_a_b @ Z2 @ T3 ) ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ Xa ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( Y
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q4 @ Xa ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q4 ) @ Xa ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q13 @ Xa ) @ ( relational_erase_a_b @ Q24 @ Xa ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ Xa ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q13 @ Xa ) @ ( relational_erase_a_b @ Q24 @ Xa ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ Xa ) ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) )
=> ( ( ( ( Xa = Z2 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
& ( ( Xa != Z2 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z2 @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.pelims
thf(fact_682_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_683_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_684_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_685_qps__union,axiom,
! [X6: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] :
( ( relational_qps_a_b @ ( sup_su5130108678486352897la_a_b @ X6 @ Y5 ) )
= ( sup_su5130108678486352897la_a_b @ ( relational_qps_a_b @ X6 ) @ ( relational_qps_a_b @ Y5 ) ) ) ).
% qps_union
thf(fact_686_qps__in,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( member4680049679412964150la_a_b @ Q @ G ) ) ).
% qps_in
thf(fact_687_finite__qps,axiom,
! [G: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ G )
=> ( finite5600759454172676150la_a_b @ ( relational_qps_a_b @ G ) ) ) ).
% finite_qps
thf(fact_688_qp__impl_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X5: nat,C4: a] :
( X
!= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) )
=> ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X5 @ Ts2 ) )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).
% qp_impl.cases
thf(fact_689_genempty_Osimps,axiom,
( relati5999705594545617851ty_a_b
= ( ^ [A3: relational_fmla_a_b] :
( ( A3
= ( relational_Bool_a_b @ $false ) )
| ? [Q5: relational_fmla_a_b] :
( ( A3
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q5 ) ) )
& ( relati5999705594545617851ty_a_b @ Q5 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A3
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q25 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A3
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q25 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A3
= ( relational_Disj_a_b @ Q14 @ Q25 ) )
& ( relati5999705594545617851ty_a_b @ Q14 )
& ( relati5999705594545617851ty_a_b @ Q25 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A3
= ( relational_Conj_a_b @ Q14 @ Q25 ) )
& ( ( relati5999705594545617851ty_a_b @ Q14 )
| ( relati5999705594545617851ty_a_b @ Q25 ) ) )
| ? [Q5: relational_fmla_a_b] :
( ? [X3: nat,Y3: nat] :
( A3
= ( relational_Conj_a_b @ Q5 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y3 ) ) ) )
& ( relati5999705594545617851ty_a_b @ Q5 ) )
| ? [Q5: relational_fmla_a_b] :
( ? [Y3: nat] :
( A3
= ( relati591517084277583526ts_a_b @ Y3 @ Q5 ) )
& ( relati5999705594545617851ty_a_b @ Q5 ) ) ) ) ) ).
% genempty.simps
thf(fact_690_genempty_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ A2 )
=> ( ( A2
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( relati5999705594545617851ty_a_b @ Q13 )
=> ~ ( relati5999705594545617851ty_a_b @ Q24 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ~ ( ( relati5999705594545617851ty_a_b @ Q13 )
| ( relati5999705594545617851ty_a_b @ Q24 ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ? [X5: nat,Y6: nat] :
( A2
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y6 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ? [Y6: nat,X5: nat] :
( A2
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y6 @ ( relational_Var_a @ X5 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
=> ~ ! [Q4: relational_fmla_a_b] :
( ? [Y6: nat] :
( A2
= ( relati591517084277583526ts_a_b @ Y6 @ Q4 ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).
% genempty.cases
thf(fact_691_member__remove,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( remove4261432235257513082la_a_b @ Y @ A ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_692_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_693_genempty_Ointros_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q1 )
=> ( ( relati5999705594545617851ty_a_b @ Q22 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(5)
thf(fact_694_genempty_Ointros_I1_J,axiom,
relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).
% genempty.intros(1)
thf(fact_695_genempty__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= ( relational_Bool_a_b @ $false ) ) ) ).
% genempty_cp
thf(fact_696_genempty_Ointros_I4_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(4)
thf(fact_697_genempty_Ointros_I3_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).
% genempty.intros(3)
thf(fact_698_qp__impl_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y )
=> ( ( ? [X5: nat,C4: a] :
( X
= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) )
=> ~ Y )
=> ( ( ? [X5: b,Ts2: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ X5 @ Ts2 ) )
=> ~ Y )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( Y
= ( ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) ) ) )
=> ( ( ? [V2: $o] :
( X
= ( relational_Bool_a_b @ V2 ) )
=> Y )
=> ( ( ? [V2: nat,Vb: nat] :
( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> Y )
=> ( ( ? [V2: relational_fmla_a_b] :
( X
= ( relational_Neg_a_b @ V2 ) )
=> Y )
=> ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> Y )
=> ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> Y ) ) ) ) ) ) ) ) ) ).
% qp_impl.elims(1)
thf(fact_699_fv__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_erase_a_b @ Q @ X ) ) @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_erase
thf(fact_700_fv__exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_exists
thf(fact_701_infinite__eval__Conj,axiom,
! [X: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Y: nat] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q @ I ) )
=> ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) @ I ) ) ) ) ).
% infinite_eval_Conj
thf(fact_702_fmla_Oset__cases_I2_J,axiom,
! [E: b,A2: relational_fmla_a_b] :
( ( member_b @ E @ ( relati8924981150291758614la_a_b @ A2 ) )
=> ( ! [Z22: list_R6823256787227418703term_a] :
( A2
!= ( relational_Pred_b_a @ E @ Z22 ) )
=> ( ! [Z2: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ Z2 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
=> ( ! [Z1: relational_fmla_a_b] :
( ? [Z22: relational_fmla_a_b] :
( A2
= ( relational_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
=> ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
( ( A2
= ( relational_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
=> ( ! [Z1: relational_fmla_a_b] :
( ? [Z22: relational_fmla_a_b] :
( A2
= ( relational_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
=> ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
( ( A2
= ( relational_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: relational_fmla_a_b] :
( ( A2
= ( relati591517084277583526ts_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) ) ) ) ) ) ) ) ) ).
% fmla.set_cases(2)
thf(fact_703_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_704_subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ( member4680049679412964150la_a_b @ X5 @ B2 ) )
=> ( ord_le4112832032246704949la_a_b @ A @ B2 ) ) ).
% subsetI
thf(fact_705_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_706_sup_Obounded__iff,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 )
= ( ( ord_le4112832032246704949la_a_b @ B @ A2 )
& ( ord_le4112832032246704949la_a_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_707_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_708_le__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( ( ord_le4112832032246704949la_a_b @ X @ Z )
& ( ord_le4112832032246704949la_a_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_709_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_710_insert__subset,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B2 )
= ( ( member4680049679412964150la_a_b @ X @ B2 )
& ( ord_le4112832032246704949la_a_b @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_711_insert__subset,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_712_Un__subset__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 )
= ( ( ord_le4112832032246704949la_a_b @ A @ C2 )
& ( ord_le4112832032246704949la_a_b @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_713_cp__exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
= ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).
% cp_exists
thf(fact_714_nocp__exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% nocp_exists
thf(fact_715_eval__Bool__False,axiom,
! [I: product_prod_b_nat > set_list_a] :
( ( relational_eval_a_b @ ( relational_Bool_a_b @ $false ) @ I )
= bot_bot_set_list_a ) ).
% eval_Bool_False
thf(fact_716_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_717_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_718_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_719_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_720_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_721_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_722_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_723_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_724_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_725_dual__order_Oeq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_726_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_727_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_728_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_729_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_730_order__subst1,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F4 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_731_order__subst2,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F4 @ B ) @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_732_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_733_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_734_ord__eq__le__subst,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( A2
= ( F4 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_735_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F4 @ B )
= C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_736_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_737_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_738_in__mono,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_739_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_740_subsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_741_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_742_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A4 )
=> ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_743_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_744_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [T6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T6 @ A4 )
=> ( member4680049679412964150la_a_b @ T6 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_745_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T6: nat] :
( ( member_nat @ T6 @ A4 )
=> ( member_nat @ T6 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_746_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P2 @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_747_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_748_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_749_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_750_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_751_fmla_Oset__intros_I13_J,axiom,
! [Yy: b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ X61 ) )
=> ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fmla.set_intros(13)
thf(fact_752_fmla_Oset__intros_I14_J,axiom,
! [Za: b,X62: relational_fmla_a_b,X61: relational_fmla_a_b] :
( ( member_b @ Za @ ( relati8924981150291758614la_a_b @ X62 ) )
=> ( member_b @ Za @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fmla.set_intros(14)
thf(fact_753_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_nat @ X5 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_754_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_nat @ A2 @ X5 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_755_subset__insert,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B2 ) )
= ( ord_le4112832032246704949la_a_b @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_756_subset__insert,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_757_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_758_rev__finite__subset,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( finite5600759454172676150la_a_b @ A ) ) ) ).
% rev_finite_subset
thf(fact_759_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_760_infinite__super,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ S @ T )
=> ( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ T ) ) ) ).
% infinite_super
thf(fact_761_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_762_finite__subset,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( finite5600759454172676150la_a_b @ B2 )
=> ( finite5600759454172676150la_a_b @ A ) ) ) ).
% finite_subset
thf(fact_763_sup_OcoboundedI2,axiom,
! [C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ B )
=> ( ord_le4112832032246704949la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_764_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_765_sup_OcoboundedI1,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ A2 )
=> ( ord_le4112832032246704949la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_766_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_767_sup_Oabsorb__iff2,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_768_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_769_sup_Oabsorb__iff1,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B3: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_770_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_771_sup_Ocobounded2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_772_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_773_sup_Ocobounded1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_774_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_775_sup_Oorder__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B3: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( A3
= ( sup_su5130108678486352897la_a_b @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_776_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_777_sup_OboundedI,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ C @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_778_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_779_sup_OboundedE,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 )
=> ~ ( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ~ ( ord_le4112832032246704949la_a_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_780_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_781_sup__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ Y )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_782_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_783_sup__absorb1,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_784_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_785_sup_Oabsorb2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_786_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_787_sup_Oabsorb1,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_788_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_789_sup__unique,axiom,
! [F4: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X5 @ ( F4 @ X5 @ Y6 ) )
=> ( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y6 @ ( F4 @ X5 @ Y6 ) )
=> ( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y6 @ X5 )
=> ( ( ord_le4112832032246704949la_a_b @ Z2 @ X5 )
=> ( ord_le4112832032246704949la_a_b @ ( F4 @ Y6 @ Z2 ) @ X5 ) ) )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= ( F4 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_790_sup__unique,axiom,
! [F4: nat > nat > nat,X: nat,Y: nat] :
( ! [X5: nat,Y6: nat] : ( ord_less_eq_nat @ X5 @ ( F4 @ X5 @ Y6 ) )
=> ( ! [X5: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ ( F4 @ X5 @ Y6 ) )
=> ( ! [X5: nat,Y6: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y6 @ X5 )
=> ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ ( F4 @ Y6 @ Z2 ) @ X5 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F4 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_791_sup_OorderI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A2
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( ord_le4112832032246704949la_a_b @ B @ A2 ) ) ).
% sup.orderI
thf(fact_792_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_793_sup_OorderE,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( A2
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_794_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_795_le__iff__sup,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [X3: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_796_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_797_sup__least,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( ord_le4112832032246704949la_a_b @ Z @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_798_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_799_sup__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ D )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_800_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_801_sup_Omono,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ D @ B )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ C @ D ) @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_802_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_803_le__supI2,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ B )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_804_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_805_le__supI1,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ A2 )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_806_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_807_sup__ge2,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_808_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_809_sup__ge1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_810_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_811_le__supI,axiom,
! [A2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ X )
=> ( ( ord_le4112832032246704949la_a_b @ B @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_812_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_813_le__supE,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ X )
=> ~ ( ( ord_le4112832032246704949la_a_b @ A2 @ X )
=> ~ ( ord_le4112832032246704949la_a_b @ B @ X ) ) ) ).
% le_supE
thf(fact_814_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_815_inf__sup__ord_I3_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_816_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_817_inf__sup__ord_I4_J,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_818_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_819_fmla_Oset__intros_I9_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a] : ( member_b @ X11 @ ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) ) ) ).
% fmla.set_intros(9)
thf(fact_820_subset__Un__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_821_subset__UnE,axiom,
! [C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
=> ~ ! [A7: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A7 @ A )
=> ! [B7: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B7 @ B2 )
=> ( C2
!= ( sup_su5130108678486352897la_a_b @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_822_Un__absorb2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_823_Un__absorb1,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_824_Un__upper2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% Un_upper2
thf(fact_825_Un__upper1,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% Un_upper1
thf(fact_826_Un__least,axiom,
! [A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ C2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_827_Un__mono,axiom,
! [A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ C2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ ( sup_su5130108678486352897la_a_b @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_828_qp__Disj,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ).
% qp_Disj
thf(fact_829_qp__impl_Osimps_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_qp_a_b @ Q ) ) ) ).
% qp_impl.simps(3)
thf(fact_830_qp__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% qp_cp_triv
thf(fact_831_qp__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q )
=> ( relational_qp_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% qp_cp
thf(fact_832_qps__qp,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( relational_qp_a_b @ Q ) ) ).
% qps_qp
thf(fact_833_finite__eval__Disj2D,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ I ) )
=> ( finite_finite_list_a @ ( relational_eval_a_b @ Q22 @ I ) ) ) ).
% finite_eval_Disj2D
thf(fact_834_qp__impl_Osimps_I8_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V @ Va ) ) ).
% qp_impl.simps(8)
thf(fact_835_qp__impl_Osimps_I4_J,axiom,
! [V: $o] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).
% qp_impl.simps(4)
thf(fact_836_qp__impl_Osimps_I2_J,axiom,
! [X: b,Ts: list_R6823256787227418703term_a] : ( relati3725921752842749053pl_a_b @ ( relational_Pred_b_a @ X @ Ts ) ) ).
% qp_impl.simps(2)
thf(fact_837_finite__csts__term,axiom,
! [T2: relational_term_nat] : ( finite_finite_nat @ ( relati694035416245573993rm_nat @ T2 ) ) ).
% finite_csts_term
thf(fact_838_finite__csts__term,axiom,
! [T2: relati112041753218324778la_a_b] : ( finite5600759454172676150la_a_b @ ( relati1926769566493843000la_a_b @ T2 ) ) ).
% finite_csts_term
thf(fact_839_fmla_Osimps_I128_J,axiom,
! [X2: $o] :
( ( relati8924981150291758614la_a_b @ ( relational_Bool_a_b @ X2 ) )
= bot_bot_set_b ) ).
% fmla.simps(128)
thf(fact_840_fmla_Osimps_I132_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
= ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X61 ) @ ( relati8924981150291758614la_a_b @ X62 ) ) ) ).
% fmla.simps(132)
thf(fact_841_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_842_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_843_subset__Diff__insert,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ X @ C2 ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B2 @ C2 ) )
& ~ ( member4680049679412964150la_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_844_subset__Diff__insert,axiom,
! [A: set_nat,B2: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_845_exists__Exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relati3989891337220013914ts_a_b @ X @ Q )
= ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ).
% exists_Exists
thf(fact_846_exists__def,axiom,
( relati3989891337220013914ts_a_b
= ( ^ [X3: nat,Q5: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X3 @ ( relational_fv_a_b @ Q5 ) ) @ ( relati591517084277583526ts_a_b @ X3 @ Q5 ) @ Q5 ) ) ) ).
% exists_def
thf(fact_847_Diff__subset__conv,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B2 ) @ C2 )
= ( ord_le4112832032246704949la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_848_Diff__partition,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( sup_su5130108678486352897la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_849_cp_Osimps_I5_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ Q ) ) ) ).
% cp.simps(5)
thf(fact_850_exists__cp__erase,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) )
= ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% exists_cp_erase
thf(fact_851_qp__ExistsE,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
=> ~ ( ( relational_qp_a_b @ Q )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).
% qp_ExistsE
thf(fact_852_qp__Exists,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ) ).
% qp_Exists
thf(fact_853_fv__cp,axiom,
! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ).
% fv_cp
thf(fact_854_fmla_Osimps_I127_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a] :
( ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) )
= ( insert_b @ X11 @ bot_bot_set_b ) ) ).
% fmla.simps(127)
thf(fact_855_finite__subset__induct_H,axiom,
! [F: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_856_finite__subset__induct_H,axiom,
! [F: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F )
=> ( ( ord_le4112832032246704949la_a_b @ F @ A )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_857_finite__subset__induct,axiom,
! [F: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_858_finite__subset__induct,axiom,
! [F: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F )
=> ( ( ord_le4112832032246704949la_a_b @ F @ A )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( member4680049679412964150la_a_b @ A6 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_859_subset__insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B2 ) )
= ( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B2 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_860_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_861_qp__impl_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X5: nat,C4: a] :
( X
!= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) )
=> ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X5 @ Ts2 ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) ) ) ) ) ).
% qp_impl.elims(2)
thf(fact_862_finite__remove__induct,axiom,
! [B2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ B2 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A5 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_863_finite__remove__induct,axiom,
! [B2: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ B2 )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [A5: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A5 )
=> ( ( A5 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A5 @ B2 )
=> ( ! [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A5 )
=> ( P2 @ ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P2 @ A5 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_864_remove__induct,axiom,
! [P2: set_nat > $o,B2: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ B2 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( P2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X7 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A5 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_865_remove__induct,axiom,
! [P2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b] :
( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ( ~ ( finite5600759454172676150la_a_b @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A5: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A5 )
=> ( ( A5 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A5 @ B2 )
=> ( ! [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ A5 )
=> ( P2 @ ( minus_4077726661957047470la_a_b @ A5 @ ( insert7010464514620295119la_a_b @ X7 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P2 @ A5 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_866_qp__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% qp_cp_erase
thf(fact_867_qp__impl_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% qp_impl.elims(3)
thf(fact_868_qp__impl_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X5: nat,C4: a] :
( ( X
= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) ) ) )
=> ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X5 @ Ts2 ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X5 @ Ts2 ) ) ) )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( Y
= ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Vb: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(1)
thf(fact_869_qp__impl_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
=> ( ! [V2: nat,Vb: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(3)
thf(fact_870_it__step__insert__iff,axiom,
! [It: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ It @ S )
=> ( ( member4680049679412964150la_a_b @ X @ It )
=> ( ( minus_4077726661957047470la_a_b @ S @ ( minus_4077726661957047470la_a_b @ It @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_871_it__step__insert__iff,axiom,
! [It: set_nat,S: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ It @ S )
=> ( ( member_nat @ X @ It )
=> ( ( minus_minus_set_nat @ S @ ( minus_minus_set_nat @ It @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_872_finite__ranking__induct,axiom,
! [S: set_nat,P2: set_nat > $o,F4: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X5: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F4 @ Y7 ) @ ( F4 @ X5 ) ) )
=> ( ( P2 @ S2 )
=> ( P2 @ ( insert_nat @ X5 @ S2 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_873_finite__ranking__induct,axiom,
! [S: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o,F4: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b,S2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S2 )
=> ( ! [Y7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F4 @ Y7 ) @ ( F4 @ X5 ) ) )
=> ( ( P2 @ S2 )
=> ( P2 @ ( insert7010464514620295119la_a_b @ X5 @ S2 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_874_qp__impl_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X5: nat,C4: a] :
( ( X
= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) ) )
=> ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X5 @ Ts2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X5 @ Ts2 ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
& ( relational_qp_a_b @ Q4 ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(2)
thf(fact_875_image__eqI,axiom,
! [B: relational_fmla_a_b,F4: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B
= ( F4 @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ B @ ( image_6790371041703824709la_a_b @ F4 @ A ) ) ) ) ).
% image_eqI
thf(fact_876_image__eqI,axiom,
! [B: nat,F4: relational_fmla_a_b > nat,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B
= ( F4 @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member_nat @ B @ ( image_341122591648980342_b_nat @ F4 @ A ) ) ) ) ).
% image_eqI
thf(fact_877_image__eqI,axiom,
! [B: relational_fmla_a_b,F4: nat > relational_fmla_a_b,X: nat,A: set_nat] :
( ( B
= ( F4 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member4680049679412964150la_a_b @ B @ ( image_4386371547000553590la_a_b @ F4 @ A ) ) ) ) ).
% image_eqI
thf(fact_878_image__eqI,axiom,
! [B: nat,F4: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F4 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F4 @ A ) ) ) ) ).
% image_eqI
thf(fact_879_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_880_finite__imageI,axiom,
! [F: set_nat,H: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ F )
=> ( finite5600759454172676150la_a_b @ ( image_4386371547000553590la_a_b @ H @ F ) ) ) ).
% finite_imageI
thf(fact_881_finite__imageI,axiom,
! [F: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ F )
=> ( finite_finite_nat @ ( image_341122591648980342_b_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_882_finite__imageI,axiom,
! [F: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ F )
=> ( finite5600759454172676150la_a_b @ ( image_6790371041703824709la_a_b @ H @ F ) ) ) ).
% finite_imageI
thf(fact_883_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: relational_fmla_a_b,F4: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( B
= ( F4 @ X ) )
=> ( member4680049679412964150la_a_b @ B @ ( image_6790371041703824709la_a_b @ F4 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_884_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: nat,F4: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( B
= ( F4 @ X ) )
=> ( member_nat @ B @ ( image_341122591648980342_b_nat @ F4 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_885_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: relational_fmla_a_b,F4: nat > relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F4 @ X ) )
=> ( member4680049679412964150la_a_b @ B @ ( image_4386371547000553590la_a_b @ F4 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_886_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F4: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F4 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F4 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_887_imageI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F4 @ X ) @ ( image_6790371041703824709la_a_b @ F4 @ A ) ) ) ).
% imageI
thf(fact_888_imageI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( member_nat @ ( F4 @ X ) @ ( image_341122591648980342_b_nat @ F4 @ A ) ) ) ).
% imageI
thf(fact_889_imageI,axiom,
! [X: nat,A: set_nat,F4: nat > relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F4 @ X ) @ ( image_4386371547000553590la_a_b @ F4 @ A ) ) ) ).
% imageI
thf(fact_890_imageI,axiom,
! [X: nat,A: set_nat,F4: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F4 @ X ) @ ( image_nat_nat @ F4 @ A ) ) ) ).
% imageI
thf(fact_891_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ( member4680049679412964150la_a_b @ ( F4 @ X5 ) @ B2 ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F4 @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_892_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > nat,B2: set_nat] :
( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ( member_nat @ ( F4 @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_341122591648980342_b_nat @ F4 @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_893_image__subsetI,axiom,
! [A: set_nat,F4: nat > relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( member4680049679412964150la_a_b @ ( F4 @ X5 ) @ B2 ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F4 @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_894_image__subsetI,axiom,
! [A: set_nat,F4: nat > nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( member_nat @ ( F4 @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_895_image__Un,axiom,
! [F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F4 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ F4 @ A ) @ ( image_6790371041703824709la_a_b @ F4 @ B2 ) ) ) ).
% image_Un
thf(fact_896_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_897_infinite__surj,axiom,
! [A: set_nat,F4: nat > nat,B2: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F4 @ B2 ) )
=> ~ ( finite_finite_nat @ B2 ) ) ) ).
% infinite_surj
thf(fact_898_infinite__surj,axiom,
! [A: set_nat,F4: relational_fmla_a_b > nat,B2: set_Re381260168593705685la_a_b] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_341122591648980342_b_nat @ F4 @ B2 ) )
=> ~ ( finite5600759454172676150la_a_b @ B2 ) ) ) ).
% infinite_surj
thf(fact_899_infinite__surj,axiom,
! [A: set_Re381260168593705685la_a_b,F4: nat > relational_fmla_a_b,B2: set_nat] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( image_4386371547000553590la_a_b @ F4 @ B2 ) )
=> ~ ( finite_finite_nat @ B2 ) ) ) ).
% infinite_surj
thf(fact_900_infinite__surj,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( image_6790371041703824709la_a_b @ F4 @ B2 ) )
=> ~ ( finite5600759454172676150la_a_b @ B2 ) ) ) ).
% infinite_surj
thf(fact_901_finite__surj,axiom,
! [A: set_nat,B2: set_nat,F4: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F4 @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_902_finite__surj,axiom,
! [A: set_nat,B2: set_Re381260168593705685la_a_b,F4: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ A )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
=> ( finite5600759454172676150la_a_b @ B2 ) ) ) ).
% finite_surj
thf(fact_903_finite__surj,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_nat,F4: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_341122591648980342_b_nat @ F4 @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_904_finite__surj,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
=> ( finite5600759454172676150la_a_b @ B2 ) ) ) ).
% finite_surj
thf(fact_905_finite__subset__image,axiom,
! [B2: set_nat,F4: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F4 @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_nat @ F4 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_906_finite__subset__image,axiom,
! [B2: set_nat,F4: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_341122591648980342_b_nat @ F4 @ A ) )
=> ? [C5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C5 @ A )
& ( finite5600759454172676150la_a_b @ C5 )
& ( B2
= ( image_341122591648980342_b_nat @ F4 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_907_finite__subset__image,axiom,
! [B2: set_Re381260168593705685la_a_b,F4: nat > relational_fmla_a_b,A: set_nat] :
( ( finite5600759454172676150la_a_b @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_4386371547000553590la_a_b @ F4 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_908_finite__subset__image,axiom,
! [B2: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
=> ? [C5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C5 @ A )
& ( finite5600759454172676150la_a_b @ C5 )
& ( B2
= ( image_6790371041703824709la_a_b @ F4 @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_909_ex__finite__subset__image,axiom,
! [F4: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F4 @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P2 @ ( image_nat_nat @ F4 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_910_ex__finite__subset__image,axiom,
! [F4: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b,P2: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_341122591648980342_b_nat @ F4 @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ A )
& ( P2 @ ( image_341122591648980342_b_nat @ F4 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_911_ex__finite__subset__image,axiom,
! [F4: nat > relational_fmla_a_b,A: set_nat,P2: set_Re381260168593705685la_a_b > $o] :
( ( ? [B4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P2 @ ( image_4386371547000553590la_a_b @ F4 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_912_ex__finite__subset__image,axiom,
! [F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( ? [B4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ A )
& ( P2 @ ( image_6790371041703824709la_a_b @ F4 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_913_all__finite__subset__image,axiom,
! [F4: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F4 @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P2 @ ( image_nat_nat @ F4 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_914_all__finite__subset__image,axiom,
! [F4: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_341122591648980342_b_nat @ F4 @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ A ) )
=> ( P2 @ ( image_341122591648980342_b_nat @ F4 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_915_all__finite__subset__image,axiom,
! [F4: nat > relational_fmla_a_b,A: set_nat,P2: set_Re381260168593705685la_a_b > $o] :
( ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ ( image_4386371547000553590la_a_b @ F4 @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P2 @ ( image_4386371547000553590la_a_b @ F4 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_916_all__finite__subset__image,axiom,
! [F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ ( image_6790371041703824709la_a_b @ F4 @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ A ) )
=> ( P2 @ ( image_6790371041703824709la_a_b @ F4 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_917_arg__min__least,axiom,
! [S: set_nat,Y: nat,F4: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F4 @ ( lattic7446932960582359483at_nat @ F4 @ S ) ) @ ( F4 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_918_arg__min__least,axiom,
! [S: set_Re381260168593705685la_a_b,Y: relational_fmla_a_b,F4: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ( ( member4680049679412964150la_a_b @ Y @ S )
=> ( ord_less_eq_nat @ ( F4 @ ( lattic5380700691367270794_b_nat @ F4 @ S ) ) @ ( F4 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_919_insert__subsetI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X6: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( ord_le4112832032246704949la_a_b @ X6 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_920_insert__subsetI,axiom,
! [X: nat,A: set_nat,X6: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_921_subset__emptyI,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [X5: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X5 @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% subset_emptyI
thf(fact_922_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X5: nat] :
~ ( member_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_923_in__image__insert__iff,axiom,
! [B2: set_se6865892389300016395la_a_b,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ! [C5: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C5 @ B2 )
=> ~ ( member4680049679412964150la_a_b @ X @ C5 ) )
=> ( ( member3481406638322139244la_a_b @ A @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X ) @ B2 ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_924_in__image__insert__iff,axiom,
! [B2: set_set_nat,X: nat,A: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B2 )
=> ~ ( member_nat @ X @ C5 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B2 ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_925_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_926_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_equiv_a_b @ Q @ Q2 )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_927_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_928_Sup__fin_Oinsert__remove,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) )
= X ) )
& ( ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
!= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_929_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_930_Sup__fin_Oremove,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( member3481406638322139244la_a_b @ X @ A )
=> ( ( ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ A )
= X ) )
& ( ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
!= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ A )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_931_Sup__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_932_Sup__fin_Oinsert,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_933_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_934_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_935_Sup__fin_Oin__idem,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( member3481406638322139244la_a_b @ X @ A )
=> ( ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A ) )
= ( lattic7150925611526040158la_a_b @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_936_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_937_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ord_less_eq_nat @ A6 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_938_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_939_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_940_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X5: nat,Y6: nat] :
( ( H @ ( sup_sup_nat @ X5 @ Y6 ) )
= ( sup_sup_nat @ ( H @ X5 ) @ ( H @ Y6 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N3 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_941_Sup__fin_Ohom__commute,axiom,
! [H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,N3: set_se6865892389300016395la_a_b] :
( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] :
( ( H @ ( sup_su5130108678486352897la_a_b @ X5 @ Y6 ) )
= ( sup_su5130108678486352897la_a_b @ ( H @ X5 ) @ ( H @ Y6 ) ) )
=> ( ( finite5238674622262875500la_a_b @ N3 )
=> ( ( N3 != bot_bo2891247006866115487la_a_b )
=> ( ( H @ ( lattic7150925611526040158la_a_b @ N3 ) )
= ( lattic7150925611526040158la_a_b @ ( image_7051608999182166449la_a_b @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_942_Sup__fin_Osubset,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B2 ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_943_Sup__fin_Osubset,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( B2 != bot_bo2891247006866115487la_a_b )
=> ( ( ord_le1577343677690852715la_a_b @ B2 @ A )
=> ( ( sup_su5130108678486352897la_a_b @ ( lattic7150925611526040158la_a_b @ B2 ) @ ( lattic7150925611526040158la_a_b @ A ) )
= ( lattic7150925611526040158la_a_b @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_944_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_945_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_946_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X5: nat,Y6: nat] : ( member_nat @ ( sup_sup_nat @ X5 @ Y6 ) @ ( insert_nat @ X5 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_947_Sup__fin_Oclosed,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ ( sup_su5130108678486352897la_a_b @ X5 @ Y6 ) @ ( insert2023870700798818565la_a_b @ X5 @ ( insert2023870700798818565la_a_b @ Y6 @ bot_bo2891247006866115487la_a_b ) ) )
=> ( member3481406638322139244la_a_b @ ( lattic7150925611526040158la_a_b @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_948_Sup__fin_Ounion,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_949_Sup__fin_Ounion,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ( finite5238674622262875500la_a_b @ B2 )
=> ( ( B2 != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ ( lattic7150925611526040158la_a_b @ A ) @ ( lattic7150925611526040158la_a_b @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_950_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_equiv_a_b @ Q @ Q2 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q2 ) @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_951_eval__on__False,axiom,
! [X6: set_nat,I: product_prod_b_nat > set_list_a] :
( ( relati8814510239606734169on_a_b @ X6 @ ( relational_Bool_a_b @ $false ) @ I )
= bot_bot_set_list_a ) ).
% eval_on_False
thf(fact_952_eval__on__cp,axiom,
! [X6: set_nat,Q: relational_fmla_a_b] :
( ( relati8814510239606734169on_a_b @ X6 @ ( relational_cp_a_b @ Q ) )
= ( relati8814510239606734169on_a_b @ X6 @ Q ) ) ).
% eval_on_cp
thf(fact_953_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q5: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q5 ) @ Q5 ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_954_finite__eval__on__Disj2D,axiom,
! [X6: set_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X6 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ I ) )
=> ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X6 @ Q22 @ I ) ) ) ) ).
% finite_eval_on_Disj2D
thf(fact_955_cov_H_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov'.Exists
thf(fact_956_gen__Bool__True,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
~ ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $true ) @ G ) ).
% gen_Bool_True
thf(fact_957_gen__Bool__False,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ G )
= ( G = bot_bo4495933725496725865la_a_b ) ) ).
% gen_Bool_False
thf(fact_958_gen__Pred,axiom,
! [Z: nat,P: b,Ts: list_R6823256787227418703term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ ( relational_Pred_b_a @ P @ Ts ) @ G )
= ( ( member_nat @ Z @ ( relati4569515538964159125_set_a @ Ts ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen_Pred
thf(fact_959_gen__finite,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% gen_finite
thf(fact_960_gen__qp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen_qp
thf(fact_961_gen__Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).
% gen_Gen_cp
thf(fact_962_Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).
% Gen_cp
thf(fact_963_gen_Ointros_I1_J,axiom,
! [X: nat] : ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen.intros(1)
thf(fact_964_gen__fv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% gen_fv
thf(fact_965_gen_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b @ X @ Q22 @ G22 )
=> ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen.intros(6)
thf(fact_966_qp__Gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_1 ) ) ) ).
% qp_Gen
thf(fact_967_cov_H_Ononfree,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b2 @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).
% cov'.nonfree
thf(fact_968_cov_H_ODisj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q22 @ G22 )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov'.Disj
thf(fact_969_cov_H_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q22 @ G22 )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov'.Conj
thf(fact_970_gen_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen.intros(5)
thf(fact_971_gen_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen.intros(4)
thf(fact_972_gen__empty__cp,axiom,
! [Z: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ Q @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( ( relational_cp_a_b @ Q )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_empty_cp
thf(fact_973_gen__cp__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Qqp @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_cp_erase
thf(fact_974_qp__gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% qp_gen
thf(fact_975_cov_H_ODisjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov'.DisjL
thf(fact_976_cov_H_ODisjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q22 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov'.DisjR
thf(fact_977_cov_H_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov'.ConjL
thf(fact_978_cov_H_OConjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q22 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov'.ConjR
thf(fact_979_nongens__cp,axiom,
! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relati62690040636126068ns_a_b @ ( relational_cp_a_b @ Q ) ) @ ( relati62690040636126068ns_a_b @ Q ) ) ).
% nongens_cp
thf(fact_980_sat__cp,axiom,
! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_cp_a_b @ Q ) @ I @ Sigma )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ).
% sat_cp
thf(fact_981_rrb__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q22 ) ) ) ).
% rrb_simps(5)
thf(fact_982_rrb__simps_I1_J,axiom,
! [B: $o] : ( relational_rrb_a_b @ ( relational_Bool_a_b @ B ) ) ).
% rrb_simps(1)
thf(fact_983_rrb__simps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] : ( relational_rrb_a_b @ ( relational_Pred_b_a @ P @ Ts ) ) ).
% rrb_simps(2)
thf(fact_984_rrb__simps_I8_J,axiom,
! [Y: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y @ Qy ) )
= ( ( ( member_nat @ Y @ ( relational_fv_a_b @ Qy ) )
=> ? [X8: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy @ X8 ) )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(8)
thf(fact_985_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N4: nat] :
( ( member_nat @ N4 @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N4 )
= ( Sigma2 @ N4 ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_986_cov__finite,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% cov_finite
thf(fact_987_rrb__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_rrb_a_b @ Q )
=> ( relational_rrb_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% rrb_cp
thf(fact_988_ex__cov,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_rrb_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_cov_a_b @ X @ Q @ X_1 ) ) ) ).
% ex_cov
thf(fact_989_sat_Osimps_I2_J,axiom,
! [B: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Bool_a_b @ B ) @ I @ Sigma )
= B ) ).
% sat.simps(2)
thf(fact_990_sat_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
| ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(6)
thf(fact_991_qps__rrb,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( relational_rrb_a_b @ Q ) ) ).
% qps_rrb
thf(fact_992_sat__fun__upd,axiom,
! [N5: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
( ~ ( member_nat @ N5 @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N5 @ Z ) )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).
% sat_fun_upd
thf(fact_993_eval__cong,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q2 ) )
=> ( ! [Sigma3: nat > a] :
( ( relational_sat_a_b @ Q @ I @ Sigma3 )
= ( relational_sat_a_b @ Q2 @ I @ Sigma3 ) )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ).
% eval_cong
thf(fact_994_cov_Ononfree,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).
% cov.nonfree
thf(fact_995_cov__fv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% cov_fv
thf(fact_996_cov_ODisj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q22 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Disj
thf(fact_997_cov_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q22 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Conj
thf(fact_998_rrb__def,axiom,
( relational_rrb_a_b
= ( ^ [Q5: relational_fmla_a_b] :
! [Y3: nat,Qy2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Qy2 ) @ ( relational_sub_a_b @ Q5 ) )
=> ? [X8: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y3 @ Qy2 @ X8 ) ) ) ) ).
% rrb_def
thf(fact_999_cov_ODisjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q22 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov.DisjR
thf(fact_1000_cov_ODisjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov.DisjL
thf(fact_1001_cov_OConjR,axiom,
! [X: nat,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q22 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q22 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov.ConjR
thf(fact_1002_cov_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ) ).
% cov.ConjL
thf(fact_1003_infinite__Implies__mono__on,axiom,
! [X6: set_nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Q2: relational_fmla_a_b] :
( ~ ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X6 @ Q @ I ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ! [X_1: nat > a] : ( relational_sat_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q ) @ Q2 ) @ I @ X_1 )
=> ~ ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X6 @ Q2 @ I ) ) ) ) ) ).
% infinite_Implies_mono_on
thf(fact_1004_cov_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov.Exists
thf(fact_1005_cov__fv__aux,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% cov_fv_aux
thf(fact_1006_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1007_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1008_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1009_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1010_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1011_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1012_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1013_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1014_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1015_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1016_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1017_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1018_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1019_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1020_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1021_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1022_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1023_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1024_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1025_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1026_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1027_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1028_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1029_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1030_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1031_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1032_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1033_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1034_order__le__less__subst1,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F4 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1035_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F4 @ B ) @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1036_order__less__le__subst1,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F4 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1037_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F4 @ B ) @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1038_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1039_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1040_psubsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B2 )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1041_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1042_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1043_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1044_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1045_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1046_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_1047_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1048_order__less__subst2,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F4 @ B ) @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1049_order__less__subst1,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F4 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1050_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1051_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F4: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F4 @ B )
= C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1052_ord__eq__less__subst,axiom,
! [A2: nat,F4: nat > nat,B: nat,C: nat] :
( ( A2
= ( F4 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
=> ( ord_less_nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
=> ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1053_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_1054_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_1055_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1056_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1057_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1058_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1059_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1060_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1061_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1062_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_1063_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P2 @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1064_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X9: nat] : ( P4 @ X9 ) )
= ( ^ [P5: nat > $o] :
? [N: nat] :
( ( P5 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P5 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_1065_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1066_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_1067_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1068_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1069_less__induct,axiom,
! [P2: nat > $o,A2: nat] :
( ! [X5: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X5 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X5 ) )
=> ( P2 @ A2 ) ) ).
% less_induct
thf(fact_1070_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1071_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1072_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_1073_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1074_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1075_psubset__imp__ex__mem,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B2 )
=> ? [B6: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ B6 @ ( minus_4077726661957047470la_a_b @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1076_psubset__imp__ex__mem,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1077_sup_Ostrict__coboundedI2,axiom,
! [C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ C @ B )
=> ( ord_le7152733262289451305la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1078_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1079_sup_Ostrict__coboundedI1,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ C @ A2 )
=> ( ord_le7152733262289451305la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1080_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1081_sup_Ostrict__order__iff,axiom,
( ord_le7152733262289451305la_a_b
= ( ^ [B3: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( ( A3
= ( sup_su5130108678486352897la_a_b @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1082_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1083_sup_Ostrict__boundedE,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 )
=> ~ ( ( ord_le7152733262289451305la_a_b @ B @ A2 )
=> ~ ( ord_le7152733262289451305la_a_b @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1084_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1085_sup_Oabsorb4,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1086_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1087_sup_Oabsorb3,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ B @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_1088_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_1089_less__supI2,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ X @ B )
=> ( ord_le7152733262289451305la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_1090_less__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_1091_less__supI1,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ X @ A2 )
=> ( ord_le7152733262289451305la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_1092_less__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_1093_finite__psubset__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A5 )
=> ( P2 @ B8 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_1094_finite__psubset__induct,axiom,
! [A: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ! [A5: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A5 )
=> ( ! [B8: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ B8 @ A5 )
=> ( P2 @ B8 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_1095_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1096_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1097_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X5 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1098_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X6 )
& ( ord_less_nat @ X5 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_1099_finite__linorder__max__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B6: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( ord_less_nat @ X7 @ B6 ) )
=> ( ( P2 @ A5 )
=> ( P2 @ ( insert_nat @ B6 @ A5 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1100_finite__linorder__min__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B6: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( ord_less_nat @ B6 @ X7 ) )
=> ( ( P2 @ A5 )
=> ( P2 @ ( insert_nat @ B6 @ A5 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1101_remove__subset,axiom,
! [X: relational_fmla_a_b,S: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ S )
=> ( ord_le7152733262289451305la_a_b @ ( minus_4077726661957047470la_a_b @ S @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ S ) ) ).
% remove_subset
thf(fact_1102_remove__subset,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_1103_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F4: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X7: nat] :
( ( member_nat @ X7 @ S )
& ( ord_less_nat @ ( F4 @ X7 ) @ ( F4 @ ( lattic7446932960582359483at_nat @ F4 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1104_arg__min__if__finite_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ S )
& ( ord_less_nat @ ( F4 @ X7 ) @ ( F4 @ ( lattic5380700691367270794_b_nat @ F4 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1105_psubset__insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B2 ) )
= ( ( ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ord_le7152733262289451305la_a_b @ A @ B2 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B2 )
=> ( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le7152733262289451305la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B2 ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1106_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ B2 )
=> ( ord_less_set_nat @ A @ B2 ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1107_finite__induct__select,axiom,
! [S: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [T7: set_nat] :
( ( ord_less_set_nat @ T7 @ S )
=> ( ( P2 @ T7 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ ( minus_minus_set_nat @ S @ T7 ) )
& ( P2 @ ( insert_nat @ X7 @ T7 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1108_finite__induct__select,axiom,
! [S: set_Re381260168593705685la_a_b,P2: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P2 @ bot_bo4495933725496725865la_a_b )
=> ( ! [T7: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ T7 @ S )
=> ( ( P2 @ T7 )
=> ? [X7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X7 @ ( minus_4077726661957047470la_a_b @ S @ T7 ) )
& ( P2 @ ( insert7010464514620295119la_a_b @ X7 @ T7 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1109_qp__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_qp_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% qp_fresh_val
thf(fact_1110_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1111_finite__Collect__conjI,axiom,
! [P2: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P2 ) )
| ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q ) ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1112_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1113_finite__Collect__disjI,axiom,
! [P2: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P2 ) )
& ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1114_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1115_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1116_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1117_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1118_finite__Collect__subsets,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( finite5238674622262875500la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [B4: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1119_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1120_finite__less__ub,axiom,
! [F4: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F4 @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F4 @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1121_less__eq__set__def,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ord_le7191224889845164944_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A4 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1122_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1123_Collect__subset,axiom,
! [A: set_Re381260168593705685la_a_b,P2: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1124_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1125_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A4: set_nat] :
( collect_set_nat
@ ^ [X8: set_nat] :
( ( ord_less_eq_set_nat @ X8 @ A4 )
& ( finite_finite_nat @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_1126_Fpow__def,axiom,
( finite3079993003003454393la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X8: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X8 @ A4 )
& ( finite5600759454172676150la_a_b @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_1127_less__set__def,axiom,
( ord_le7152733262289451305la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ord_le6021219098528097948_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A4 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_1128_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_1129_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1130_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K @ M2 )
=> ? [N6: nat] :
( ( ord_less_nat @ M2 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1131_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P2 @ K2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1132_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1133_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N5: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ N3 )
=> ( ord_less_nat @ X5 @ N5 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1134_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
? [N: nat,F2: nat > nat] :
( A4
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1135_finite__conv__nat__seg__image,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] :
? [N: nat,F2: nat > relational_fmla_a_b] :
( A4
= ( image_4386371547000553590la_a_b @ F2
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1136_nat__seg__image__imp__finite,axiom,
! [A: set_nat,F4: nat > nat,N5: nat] :
( ( A
= ( image_nat_nat @ F4
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N5 ) ) ) )
=> ( finite_finite_nat @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1137_nat__seg__image__imp__finite,axiom,
! [A: set_Re381260168593705685la_a_b,F4: nat > relational_fmla_a_b,N5: nat] :
( ( A
= ( image_4386371547000553590la_a_b @ F4
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N5 ) ) ) )
=> ( finite5600759454172676150la_a_b @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1138_Collect__conv__if,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P2 @ X3 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P2 @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1139_Collect__conv__if2,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P2 @ X3 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P2 @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1140_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% Set.empty_def
thf(fact_1141_pigeonhole__infinite,axiom,
! [A: set_nat,F4: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F4 @ A ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( ( F4 @ A3 )
= ( F4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1142_pigeonhole__infinite,axiom,
! [A: set_nat,F4: nat > relational_fmla_a_b] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite5600759454172676150la_a_b @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( ( F4 @ A3 )
= ( F4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1143_pigeonhole__infinite,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > nat] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ ( image_341122591648980342_b_nat @ F4 @ A ) )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A )
& ( ( F4 @ A3 )
= ( F4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1144_pigeonhole__infinite,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > relational_fmla_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite5600759454172676150la_a_b @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A )
& ( ( F4 @ A3 )
= ( F4 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1145_Compr__image__eq,axiom,
! [F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P2: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
& ( P2 @ X3 ) ) )
= ( image_6790371041703824709la_a_b @ F4
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P2 @ ( F4 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1146_Compr__image__eq,axiom,
! [F4: nat > relational_fmla_a_b,A: set_nat,P2: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
& ( P2 @ X3 ) ) )
= ( image_4386371547000553590la_a_b @ F4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F4 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1147_Compr__image__eq,axiom,
! [F4: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b,P2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_341122591648980342_b_nat @ F4 @ A ) )
& ( P2 @ X3 ) ) )
= ( image_341122591648980342_b_nat @ F4
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P2 @ ( F4 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1148_Compr__image__eq,axiom,
! [F4: nat > nat,A: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F4 @ A ) )
& ( P2 @ X3 ) ) )
= ( image_nat_nat @ F4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F4 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1149_imageE,axiom,
! [B: relational_fmla_a_b,F4: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B @ ( image_6790371041703824709la_a_b @ F4 @ A ) )
=> ~ ! [X5: relational_fmla_a_b] :
( ( B
= ( F4 @ X5 ) )
=> ~ ( member4680049679412964150la_a_b @ X5 @ A ) ) ) ).
% imageE
thf(fact_1150_imageE,axiom,
! [B: relational_fmla_a_b,F4: nat > relational_fmla_a_b,A: set_nat] :
( ( member4680049679412964150la_a_b @ B @ ( image_4386371547000553590la_a_b @ F4 @ A ) )
=> ~ ! [X5: nat] :
( ( B
= ( F4 @ X5 ) )
=> ~ ( member_nat @ X5 @ A ) ) ) ).
% imageE
thf(fact_1151_imageE,axiom,
! [B: nat,F4: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( member_nat @ B @ ( image_341122591648980342_b_nat @ F4 @ A ) )
=> ~ ! [X5: relational_fmla_a_b] :
( ( B
= ( F4 @ X5 ) )
=> ~ ( member4680049679412964150la_a_b @ X5 @ A ) ) ) ).
% imageE
thf(fact_1152_imageE,axiom,
! [B: nat,F4: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F4 @ A ) )
=> ~ ! [X5: nat] :
( ( B
= ( F4 @ X5 ) )
=> ~ ( member_nat @ X5 @ A ) ) ) ).
% imageE
thf(fact_1153_qps__def,axiom,
( relational_qps_a_b
= ( ^ [G3: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [Q5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q5 @ G3 )
& ( relational_qp_a_b @ Q5 ) ) ) ) ) ).
% qps_def
thf(fact_1154_subst_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X @ Y )
= ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q22 @ X @ Y ) ) ) ).
% subst.simps(6)
thf(fact_1155_subst_Osimps_I1_J,axiom,
! [T2: $o,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Bool_a_b @ T2 ) @ X @ Y )
= ( relational_Bool_a_b @ T2 ) ) ).
% subst.simps(1)
thf(fact_1156_eqs__def,axiom,
( relational_eqs_a_b
= ( ^ [X3: nat,G3: set_Re381260168593705685la_a_b] :
( collect_nat
@ ^ [Y3: nat] :
( ( X3 != Y3 )
& ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y3 ) ) @ G3 ) ) ) ) ) ).
% eqs_def
thf(fact_1157_set__diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A4 )
& ~ ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1158_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ~ ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1159_minus__set__def,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( minus_9215201808853403479_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A4 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1160_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1161_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_1162_insert__def,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b] :
( sup_su5130108678486352897la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_1163_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
| ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1164_Un__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A4 )
| ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1165_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1166_sup__set__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A4 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1167_Collect__disj__eq,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1168_Collect__disj__eq,axiom,
! [P2: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( collec3419995626248312948la_a_b @ P2 ) @ ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1169_insert__Collect,axiom,
! [A2: nat,P2: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1170_insert__compr,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( X3 = A3 )
| ( member4680049679412964150la_a_b @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1171_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
| ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1172_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1173_sup__Un__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1174_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1175_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_Re381260168593705685la_a_b,R: nat > relational_fmla_a_b > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite5600759454172676150la_a_b @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [Xa2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa2 @ B2 )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1176_pigeonhole__infinite__rel,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_nat,R: relational_fmla_a_b > nat > $o] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B2 )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A )
& ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1177_pigeonhole__infinite__rel,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,R: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite5600759454172676150la_a_b @ B2 )
=> ( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ? [Xa2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa2 @ B2 )
& ( R @ X5 @ Xa2 ) ) )
=> ? [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ B2 )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A )
& ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1178_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1179_not__finite__existsD,axiom,
! [P2: relational_fmla_a_b > $o] :
( ~ ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P2 ) )
=> ? [X_1: relational_fmla_a_b] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1180_finite__if__eq__beyond__finite,axiom,
! [S: set_nat,S3: set_nat] :
( ( finite_finite_nat @ S )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [S4: set_nat] :
( ( minus_minus_set_nat @ S4 @ S )
= ( minus_minus_set_nat @ S3 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_1181_finite__if__eq__beyond__finite,axiom,
! [S: set_Re381260168593705685la_a_b,S3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S )
=> ( finite5238674622262875500la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [S4: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ S4 @ S )
= ( minus_4077726661957047470la_a_b @ S3 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_1182_gen_Ointros_I8_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(8)
thf(fact_1183_gen_Ointros_I9_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(9)
thf(fact_1184_Gen__cp__subst,axiom,
! [Z: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z @ Q @ X_12 )
=> ( ( Z != X )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) @ X_1 ) ) ) ).
% Gen_cp_subst
thf(fact_1185_exists__cp__subst,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b] :
( ( X != Y )
=> ( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) )
= ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).
% exists_cp_subst
thf(fact_1186_qp__cp__subst__triv,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( relational_subst_a_b @ Q @ X @ Y ) ) ) ).
% qp_cp_subst_triv
thf(fact_1187_rrb__cp__subst,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_rrb_a_b @ Q )
=> ( relational_rrb_a_b @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).
% rrb_cp_subst
thf(fact_1188_csts_Osimps_I1_J,axiom,
! [B: $o] :
( ( relational_csts_a_b @ ( relational_Bool_a_b @ B ) )
= bot_bot_set_a ) ).
% csts.simps(1)
thf(fact_1189_csts_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q22 ) ) ) ).
% csts.simps(6)
thf(fact_1190_gen__csts,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% gen_csts
thf(fact_1191_cov__csts,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% cov_csts
thf(fact_1192_cov__nocp__intros,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_subst_a_b @ Q5 @ Y @ X )
@ Gy ) ) ) ) ) ) ).
% cov_nocp_intros
thf(fact_1193_cov_H_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_subst_a_b @ Q5 @ Y @ X )
@ Gy ) ) ) ) ) ) ).
% cov'.Exists_gen
thf(fact_1194_cov_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov.Exists_gen
thf(fact_1195_cov_H__cp__intros,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov'_cp_intros
thf(fact_1196_Eq__eq__subst__iff,axiom,
! [Y: nat,Z: nat,Q: relational_fmla_a_b,X: nat] :
( ( ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z ) )
= ( relational_subst_a_b @ Q @ X @ Y ) )
= ( ( ( Z = X )
=> ( ( X = Y )
& ( Q
= ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) ) ) )
& ( ( Z != X )
=> ( ( Q
= ( relational_Eq_a_b @ X @ ( relational_Var_a @ Z ) ) )
| ( Q
= ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z ) ) )
| ( ( Z = Y )
& ( member4680049679412964150la_a_b @ Q @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ).
% Eq_eq_subst_iff
thf(fact_1197_fv__subst,axiom,
! [X: nat,Q: relational_fmla_a_b,Y: nat] :
( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( insert_nat @ Y @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) )
& ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( relational_fv_a_b @ Q ) ) ) ) ).
% fv_subst
thf(fact_1198_ap__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% ap_cp_triv
thf(fact_1199_ap__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q )
=> ( relational_ap_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% ap_cp
thf(fact_1200_Pred,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] : ( relational_ap_a_b @ ( relational_Pred_b_a @ P @ Ts ) ) ).
% Pred
thf(fact_1201_ap__cp__subst__triv,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( relational_subst_a_b @ Q @ X @ Y ) ) ) ).
% ap_cp_subst_triv
thf(fact_1202_sat_Ocases,axiom,
! [X: produc1132964494702330949_nat_a] :
( ! [R2: b,Ts2: list_R6823256787227418703term_a,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ( ! [B6: $o,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B6 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ( ! [X5: nat,T4: relational_term_a,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X5 @ T4 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) )
=> ~ ! [Z2: nat,Phi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma3 ) ) ) ) ) ) ) ) ) ).
% sat.cases
thf(fact_1203_subst_Ocases,axiom,
! [X: produc8867654947514737559at_nat] :
( ! [T3: $o,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ( ! [Z2: nat,T3: relational_term_a,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ( ! [Q4: relational_fmla_a_b,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b,X5: nat,Y6: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) ) ) ) ) ) ) ) ).
% subst.cases
thf(fact_1204_ap_Osimps,axiom,
( relational_ap_a_b
= ( ^ [A3: relational_fmla_a_b] :
( ? [P6: b,Ts3: list_R6823256787227418703term_a] :
( A3
= ( relational_Pred_b_a @ P6 @ Ts3 ) )
| ? [X3: nat,C6: a] :
( A3
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C6 ) ) ) ) ) ) ).
% ap.simps
thf(fact_1205_ap_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relational_ap_a_b @ A2 )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( A2
!= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ~ ! [X5: nat,C4: a] :
( A2
!= ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C4 ) ) ) ) ) ).
% ap.cases
thf(fact_1206_ap__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% ap_cp_erase
thf(fact_1207_gen_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen.intros(2)
thf(fact_1208_cov_Oap,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov.ap
thf(fact_1209_cov_H_Oap,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov'.ap
thf(fact_1210_ap__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_ap_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% ap_fresh_val
thf(fact_1211_gen__induct,axiom,
! [X1: nat,X2: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
( ( relational_gen_a_b @ X1 @ X2 @ X33 )
=> ( ! [X5: nat] : ( P2 @ X5 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
=> ( ! [Q4: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q4 )
=> ! [X5: nat] :
( ( member_nat @ X5 @ ( relational_fv_a_b @ Q4 ) )
=> ( P2 @ X5 @ Q4 @ ( insert7010464514620295119la_a_b @ Q4 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ! [X5: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ Q4 @ G4 )
=> ( ( P2 @ X5 @ Q4 @ G4 )
=> ( P2 @ X5 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) @ G4 ) ) )
=> ( ! [X5: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
=> ( ( P2 @ X5 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
=> ( P2 @ X5 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
=> ( ! [X5: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
=> ( ( P2 @ X5 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
=> ( P2 @ X5 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
=> ( ! [X5: nat,Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ Q13 @ G12 )
=> ( ( P2 @ X5 @ Q13 @ G12 )
=> ! [Q24: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ Q24 @ G23 )
=> ( ( P2 @ X5 @ Q24 @ G23 )
=> ( P2 @ X5 @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) ) ) ) ) )
=> ( ! [X5: nat,Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( ( ( relational_gen_a_b @ X5 @ Q13 @ G4 )
& ( P2 @ X5 @ Q13 @ G4 ) )
| ( ( relational_gen_a_b @ X5 @ Q24 @ G4 )
& ( P2 @ X5 @ Q24 @ G4 ) ) )
=> ( P2 @ X5 @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ G4 ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y6 @ Q4 @ G4 )
=> ( ( P2 @ Y6 @ Q4 @ G4 )
=> ! [X5: nat] :
( P2 @ X5 @ ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y6 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y6 @ X5 )
@ G4 ) ) ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y6 @ Q4 @ G4 )
=> ( ( P2 @ Y6 @ Q4 @ G4 )
=> ! [X5: nat] :
( P2 @ X5 @ ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y6 @ ( relational_Var_a @ X5 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y6 @ X5 )
@ G4 ) ) ) )
=> ( ! [X5: nat,Y6: nat] :
( ( X5 != Y6 )
=> ! [Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X5 @ Q4 @ G4 )
=> ( ( P2 @ X5 @ Q4 @ G4 )
=> ( P2 @ X5 @ ( relati591517084277583526ts_a_b @ Y6 @ Q4 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y6 ) @ G4 ) ) ) ) )
=> ( P2 @ X1 @ X2 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_induct
thf(fact_1212_gen_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ Q4 @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b @ A1 @ Q13 @ G12 )
=> ~ ( relational_gen_a_b @ A1 @ Q24 @ G23 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( A32 = G4 )
=> ~ ( ( relational_gen_a_b @ A1 @ Q13 @ G4 )
| ( relational_gen_a_b @ A1 @ Q24 @ G4 ) ) ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y6 ) ) ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y6 @ A1 ) )
@ G4 ) )
=> ~ ( relational_gen_a_b @ Y6 @ Q4 @ G4 ) ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y6 @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y6 @ A1 ) )
@ G4 ) )
=> ~ ( relational_gen_a_b @ Y6 @ Q4 @ G4 ) ) )
=> ~ ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y6 @ Q4 ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y6 ) @ G4 ) )
=> ( ( A1 != Y6 )
=> ~ ( relational_gen_a_b @ A1 @ Q4 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen.cases
thf(fact_1213_gen_Osimps,axiom,
( relational_gen_a_b
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q5 ) ) )
& ( relational_gen_a_b @ A12 @ Q5 @ A33 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q25 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) @ A33 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q25 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) @ A33 ) )
| ? [Q14: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q14 @ Q25 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b @ A12 @ Q14 @ G13 )
& ( relational_gen_a_b @ A12 @ Q25 @ G24 ) ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q14 @ Q25 ) )
& ( ( relational_gen_a_b @ A12 @ Q14 @ A33 )
| ( relational_gen_a_b @ A12 @ Q25 @ A33 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q5 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y3 ) ) ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A12 ) )
@ G3 ) )
& ( relational_gen_a_b @ Y3 @ Q5 @ G3 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q5 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A12 ) )
@ G3 ) )
& ( relational_gen_a_b @ Y3 @ Q5 @ G3 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y3 @ Q5 ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G3 ) )
& ( A12 != Y3 )
& ( relational_gen_a_b @ A12 @ Q5 @ G3 ) ) ) ) ) ) ).
% gen.simps
thf(fact_1214_gen_H_Osimps,axiom,
( relational_gen_a_b2
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q5 ) ) )
& ( relational_gen_a_b2 @ A12 @ Q5 @ A33 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q25 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) @ A33 ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q25 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q25 ) ) @ A33 ) )
| ? [Q14: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q14 @ Q25 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b2 @ A12 @ Q14 @ G13 )
& ( relational_gen_a_b2 @ A12 @ Q25 @ G24 ) ) )
| ? [Q14: relational_fmla_a_b,Q25: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q14 @ Q25 ) )
& ( ( relational_gen_a_b2 @ A12 @ Q14 @ A33 )
| ( relational_gen_a_b2 @ A12 @ Q25 @ A33 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q5 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y3 ) ) ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A12 )
@ G3 ) )
& ( relational_gen_a_b2 @ Y3 @ Q5 @ G3 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q5 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A12 )
@ G3 ) )
& ( relational_gen_a_b2 @ Y3 @ Q5 @ G3 ) ) )
| ? [Y3: nat,Q5: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y3 @ Q5 ) )
& ? [G3: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G3 ) )
& ( A12 != Y3 )
& ( relational_gen_a_b2 @ A12 @ Q5 @ G3 ) ) ) ) ) ) ).
% gen'.simps
thf(fact_1215_gen_H_Ointros_I1_J,axiom,
! [X: nat] : ( relational_gen_a_b2 @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen'.intros(1)
thf(fact_1216_gen_H_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b2 @ X @ Q22 @ G22 )
=> ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen'.intros(6)
thf(fact_1217_gen_H__qp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen'_qp
thf(fact_1218_gen_H_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen'.intros(4)
thf(fact_1219_gen_H_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen'.intros(5)
thf(fact_1220_gen_H_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen'.intros(2)
thf(fact_1221_gen_H__cp__intros_I2_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(2)
thf(fact_1222_gen_H__cp__intros_I1_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q5: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q5 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(1)
thf(fact_1223_gen_H_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ Q4 @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b2 @ A1 @ Q13 @ G12 )
=> ~ ( relational_gen_a_b2 @ A1 @ Q24 @ G23 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( A32 = G4 )
=> ~ ( ( relational_gen_a_b2 @ A1 @ Q13 @ G4 )
| ( relational_gen_a_b2 @ A1 @ Q24 @ G4 ) ) ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y6 ) ) ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y6 @ A1 )
@ G4 ) )
=> ~ ( relational_gen_a_b2 @ Y6 @ Q4 @ G4 ) ) )
=> ( ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y6 @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y6 @ A1 )
@ G4 ) )
=> ~ ( relational_gen_a_b2 @ Y6 @ Q4 @ G4 ) ) )
=> ~ ! [Y6: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y6 @ Q4 ) )
=> ! [G4: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y6 ) @ G4 ) )
=> ( ( A1 != Y6 )
=> ~ ( relational_gen_a_b2 @ A1 @ Q4 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen'.cases
thf(fact_1224_cp_Osimps_I1_J,axiom,
! [X: nat,T2: relational_term_a] :
( ( relational_cp_a_b @ ( relational_Eq_a_b @ X @ T2 ) )
= ( relati582353067970734056la_a_b
@ ^ [A3: a] : ( relational_Eq_a_b @ X @ T2 )
@ ^ [Y3: nat] : ( if_Rel1279876242545935705la_a_b @ ( X = Y3 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) )
@ T2 ) ) ).
% cp.simps(1)
thf(fact_1225_fresh2_I3_J,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b] :
~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_fv_a_b @ Q ) ) ).
% fresh2(3)
thf(fact_1226_fresh2__gt_I3_J,axiom,
! [Z: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
=> ( ord_less_nat @ Z @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) ) ).
% fresh2_gt(3)
thf(fact_1227_subst__exists,axiom,
! [Z: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
=> ( ( ( X = Z )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ X @ Q ) ) )
& ( ( X != Z )
=> ( ( ( Z = Y )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
& ( ( Z != Y )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ Z @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) )
& ( ~ ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X @ Y )
= ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).
% subst_exists
thf(fact_1228_subst_Oelims,axiom,
! [X: relational_fmla_a_b,Xa: nat,Xb: nat,Y: relational_fmla_a_b] :
( ( ( relational_subst_a_b @ X @ Xa @ Xb )
= Y )
=> ( ! [T3: $o] :
( ( X
= ( relational_Bool_a_b @ T3 ) )
=> ( Y
!= ( relational_Bool_a_b @ T3 ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( Y
!= ( relational_Pred_b_a @ P3
@ ( map_Re5736185711816362116term_a
@ ^ [T6: relational_term_a] : ( relati7175845559408349773term_a @ T6 @ Xa @ Xb )
@ Ts2 ) ) ) )
=> ( ! [Z2: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z2 @ T3 ) )
=> ( Y
!= ( relational_Eq_a_b @ ( if_nat @ ( Z2 = Xa ) @ Xb @ Z2 ) @ ( relati7175845559408349773term_a @ T3 @ Xa @ Xb ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( Y
!= ( relational_Neg_a_b @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( relational_Conj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( relational_Disj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) ) )
=> ~ ! [Z2: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z2 @ Q4 ) )
=> ~ ( ( ( Xa = Z2 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
& ( ( Xa != Z2 )
=> ( ( ( Z2 = Xb )
=> ( Y
= ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q4 @ Z2 @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) ) @ Xa @ Xb ) ) ) )
& ( ( Z2 != Xb )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z2 @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% subst.elims
thf(fact_1229_finite__UN,axiom,
! [A: set_nat,B2: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite_finite_nat @ ( B2 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1230_finite__UN,axiom,
! [A: set_nat,B2: nat > set_Re381260168593705685la_a_b] :
( ( finite_finite_nat @ A )
=> ( ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ B2 @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1231_finite__UN,axiom,
! [A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b > set_nat] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ B2 @ A ) ) )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( finite_finite_nat @ ( B2 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1232_finite__UN,axiom,
! [A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B2 @ A ) ) )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1233_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M3: set_nat] :
( ( member_set_nat @ M3 @ A )
=> ( finite_finite_nat @ M3 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_1234_finite__Union,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ! [M3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ M3 @ A )
=> ( finite5600759454172676150la_a_b @ M3 ) )
=> ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ A ) ) ) ) ).
% finite_Union
thf(fact_1235_finite__UN__I,axiom,
! [A: set_nat,B2: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( finite_finite_nat @ ( B2 @ A6 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1236_finite__UN__I,axiom,
! [A: set_nat,B2: nat > set_Re381260168593705685la_a_b] :
( ( finite_finite_nat @ A )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ A6 ) ) )
=> ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ B2 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1237_finite__UN__I,axiom,
! [A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b > set_nat] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ! [A6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A6 @ A )
=> ( finite_finite_nat @ ( B2 @ A6 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ B2 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1238_finite__UN__I,axiom,
! [A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ! [A6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A6 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ A6 ) ) )
=> ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B2 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1239_finite__UNION__then__finite,axiom,
! [B2: relational_fmla_a_b > set_nat,A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ B2 @ A ) ) )
=> ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( finite_finite_nat @ ( B2 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_1240_finite__UNION__then__finite,axiom,
! [B2: nat > set_nat,A: set_nat,A2: nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( finite_finite_nat @ ( B2 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_1241_finite__UNION__then__finite,axiom,
! [B2: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B2 @ A ) ) )
=> ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_1242_finite__UNION__then__finite,axiom,
! [B2: nat > set_Re381260168593705685la_a_b,A: set_nat,A2: nat] :
( ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( finite5600759454172676150la_a_b @ ( B2 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_1243_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_1244_finite__UnionD,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5600759454172676150la_a_b @ ( comple8442120529048846632la_a_b @ A ) )
=> ( finite5238674622262875500la_a_b @ A ) ) ).
% finite_UnionD
thf(fact_1245_finite__Sup__in,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ! [X5: set_Re381260168593705685la_a_b,Y6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X5 @ A )
=> ( ( member3481406638322139244la_a_b @ Y6 @ A )
=> ( member3481406638322139244la_a_b @ ( sup_su5130108678486352897la_a_b @ X5 @ Y6 ) @ A ) ) )
=> ( member3481406638322139244la_a_b @ ( comple8442120529048846632la_a_b @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1246_subst_Osimps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Pred_b_a @ P @ Ts ) @ X @ Y )
= ( relational_Pred_b_a @ P
@ ( map_Re5736185711816362116term_a
@ ^ [T6: relational_term_a] : ( relati7175845559408349773term_a @ T6 @ X @ Y )
@ Ts ) ) ) ).
% subst.simps(2)
thf(fact_1247_Union__Un__distrib,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A ) @ ( comple8442120529048846632la_a_b @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_1248_Sup__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ A ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ A ) ) ) ).
% Sup_insert
thf(fact_1249_UN__Un,axiom,
! [M4: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ A ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ B2 ) ) ) ) ).
% UN_Un
thf(fact_1250_fv__flat__Disj,axiom,
! [Q: relational_fmla_a_b] :
( ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ relational_fv_a_b @ ( restri569617705344514291sj_a_b @ Q ) ) )
= ( relational_fv_a_b @ Q ) ) ).
% fv_flat_Disj
thf(fact_1251_Union__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ B2 ) ) ) ).
% Union_insert
thf(fact_1252_Sup__union__distrib,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A ) @ ( comple8442120529048846632la_a_b @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1253_SUP__absorb,axiom,
! [K: relational_fmla_a_b,I: set_Re381260168593705685la_a_b,A: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ K @ I )
=> ( ( sup_su5130108678486352897la_a_b @ ( A @ K ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A @ I ) ) )
= ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1254_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_Re381260168593705685la_a_b] :
( ( member_nat @ K @ I )
=> ( ( sup_su5130108678486352897la_a_b @ ( A @ K ) @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ A @ I ) ) )
= ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1255_UN__absorb,axiom,
! [K: relational_fmla_a_b,I: set_Re381260168593705685la_a_b,A: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ K @ I )
=> ( ( sup_su5130108678486352897la_a_b @ ( A @ K ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A @ I ) ) )
= ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1256_UN__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_Re381260168593705685la_a_b] :
( ( member_nat @ K @ I )
=> ( ( sup_su5130108678486352897la_a_b @ ( A @ K ) @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ A @ I ) ) )
= ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1257_SUP__union,axiom,
! [M4: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ A ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ M4 @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1258_UNION__fun__upd,axiom,
! [A: relational_fmla_a_b > set_Re381260168593705685la_a_b,I2: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b,J: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ ( fun_up452696389778429683la_a_b @ A @ I2 @ B2 ) @ J ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A @ ( minus_4077726661957047470la_a_b @ J @ ( insert7010464514620295119la_a_b @ I2 @ bot_bo4495933725496725865la_a_b ) ) ) ) @ ( if_set2835548578466827919la_a_b @ ( member4680049679412964150la_a_b @ I2 @ J ) @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_1259_UNION__fun__upd,axiom,
! [A: nat > set_Re381260168593705685la_a_b,I2: nat,B2: set_Re381260168593705685la_a_b,J: set_nat] :
( ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ ( fun_up6290740034186280484la_a_b @ A @ I2 @ B2 ) @ J ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ A @ ( minus_minus_set_nat @ J @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) ) ) @ ( if_set2835548578466827919la_a_b @ ( member_nat @ I2 @ J ) @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_1260_cSup__least,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ X5 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1261_cSup__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ X5 @ A2 ) )
=> ( ! [Y6: nat] :
( ! [X7: nat] :
( ( member_nat @ X7 @ X6 )
=> ( ord_less_eq_nat @ X7 @ Y6 ) )
=> ( ord_less_eq_nat @ A2 @ Y6 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1262_less__cSupE,axiom,
! [Y: nat,X6: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X6 ) )
=> ( ( X6 != bot_bot_set_nat )
=> ~ ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ~ ( ord_less_nat @ Y @ X5 ) ) ) ) ).
% less_cSupE
thf(fact_1263_less__cSupD,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X6 ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ X6 )
& ( ord_less_nat @ Z @ X5 ) ) ) ) ).
% less_cSupD
thf(fact_1264_le__cSup__finite,axiom,
! [X6: set_nat,X: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_1265_finite__imp__Sup__less,axiom,
! [X6: set_nat,X: nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_nat @ X5 @ A2 ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_1266_cSUP__least,axiom,
! [A: set_Re381260168593705685la_a_b,F4: relational_fmla_a_b > nat,M4: nat] :
( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X5 @ A )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ M4 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_341122591648980342_b_nat @ F4 @ A ) ) @ M4 ) ) ) ).
% cSUP_least
thf(fact_1267_cSUP__least,axiom,
! [A: set_nat,F4: nat > nat,M4: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( ord_less_eq_nat @ ( F4 @ X5 ) @ M4 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F4 @ A ) ) @ M4 ) ) ) ).
% cSUP_least
thf(fact_1268_finite__Sup__less__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ X3 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1269_cSup__eq__Sup__fin,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X6 )
= ( lattic1093996805478795353in_nat @ X6 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1270_finite__subset__Union,axiom,
! [A: set_nat,B9: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B9 ) )
=> ~ ! [F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ B9 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1271_finite__subset__Union,axiom,
! [A: set_Re381260168593705685la_a_b,B9: set_se6865892389300016395la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( comple8442120529048846632la_a_b @ B9 ) )
=> ~ ! [F5: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ F5 )
=> ( ( ord_le1577343677690852715la_a_b @ F5 @ B9 )
=> ~ ( ord_le4112832032246704949la_a_b @ A @ ( comple8442120529048846632la_a_b @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1272_csts_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_a] :
( ( ( relational_csts_a_b @ X )
= Y )
=> ( ( ? [B6: $o] :
( X
= ( relational_Bool_a_b @ B6 ) )
=> ( Y != bot_bot_set_a ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( Y
!= ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts2 ) ) ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( Y
!= ( relati6638259395341848997term_a @ T3 ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( Y
!= ( relational_csts_a_b @ Q4 ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( Y
!= ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( Y
!= ( relational_csts_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ).
% csts.elims
thf(fact_1273_csts_Osimps_I2_J,axiom,
! [P: b,Ts: list_R6823256787227418703term_a] :
( ( relational_csts_a_b @ ( relational_Pred_b_a @ P @ Ts ) )
= ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) ) ).
% csts.simps(2)
thf(fact_1274_csts_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_a] :
( ( ( relational_csts_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ X )
=> ( ! [B6: $o] :
( ( X
= ( relational_Bool_a_b @ B6 ) )
=> ( ( Y = bot_bot_set_a )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Bool_a_b @ B6 ) ) ) )
=> ( ! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts2 ) )
=> ( ( Y
= ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts2 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ) )
=> ( ! [X5: nat,T3: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X5 @ T3 ) )
=> ( ( Y
= ( relati6638259395341848997term_a @ T3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
=> ( ! [Q4: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q4 ) )
=> ( ( Y
= ( relational_csts_a_b @ Q4 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
=> ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
=> ( ( Y
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
=> ~ ! [X5: nat,Q4: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X5 @ Q4 ) )
=> ( ( Y
= ( relational_csts_a_b @ Q4 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% csts.pelims
thf(fact_1275_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1276_List_Ofinite__set,axiom,
! [Xs: list_R8263082107343818799la_a_b] : ( finite5600759454172676150la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ).
% List.finite_set
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
member4680049679412964150la_a_b @ q2 @ ( restri569617705344514291sj_a_b @ q ) ).
thf(conj_1,hypothesis,
relati1591879772219623554ed_a_b @ q ).
thf(conj_2,conjecture,
relati1591879772219623554ed_a_b @ q2 ).
%------------------------------------------------------------------------------