%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP153^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:58 AM UTC 2026
% Result : Theorem 2.20s 1.21s
% Output : Refutation 2.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 5
% Syntax : Number of formulae : 28 ( 7 unt; 0 typ; 2 def)
% Number of atoms : 99 ( 25 equ; 0 cnn)
% Maximal formula atoms : 2 ( 3 avg)
% Number of connectives : 135 ( 14 ~; 9 |; 7 &; 100 @)
% ( 2 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 87 usr; 13 con; 0-3 aty)
% Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Product_prod_a_a: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_7,type,
set_a: $tType ).
thf(type_def_9,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_10,type,
set_Pr1295299783od_a_a: $tType ).
thf(type_def_11,type,
set_set_a: $tType ).
thf(type_def_12,type,
set_Pr2045688071_set_a: $tType ).
thf(type_def_13,type,
product_prod_a_a: $tType ).
thf(type_def_14,type,
produc1691597095_set_a: $tType ).
thf(type_def_15,type,
produc1572603623od_a_a: $tType ).
thf(type_def_16,type,
produc1990712039od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1295299783od_a_a: $tType ).
thf(func_def_1,type,
produc1990712039od_a_a: $tType ).
thf(func_def_2,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_3,type,
produc1572603623od_a_a: $tType ).
thf(func_def_4,type,
set_Pr2045688071_set_a: $tType ).
thf(func_def_5,type,
produc1691597095_set_a: $tType ).
thf(func_def_6,type,
set_Product_prod_a_a: $tType ).
thf(func_def_7,type,
product_prod_a_a: $tType ).
thf(func_def_8,type,
set_set_a: $tType ).
thf(func_def_9,type,
set_a: $tType ).
thf(func_def_10,type,
a: $tType ).
thf(func_def_11,type,
bNF_Ca850758276_set_a: set_Product_prod_a_a > ( a > set_a ) > $o ).
thf(func_def_12,type,
uminus1440301054od_a_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_13,type,
uminus_uminus_set_a: set_a > set_a ).
thf(func_def_14,type,
order_653178931r_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_15,type,
order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_16,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_17,type,
order_816746613_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_18,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_19,type,
order_under_a: set_Product_prod_a_a > a > set_a ).
thf(func_def_20,type,
bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
thf(func_def_21,type,
bot_bot_a_o: a > $o ).
thf(func_def_22,type,
bot_bo1519632275_set_a: produc1691597095_set_a ).
thf(func_def_23,type,
bot_bo1116205875od_a_a: set_Pr1948701895od_a_a ).
thf(func_def_24,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_25,type,
bot_bot_set_set_a: set_set_a ).
thf(func_def_26,type,
bot_bot_set_a: set_a ).
thf(func_def_27,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(func_def_28,type,
ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_29,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_30,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_31,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_32,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(func_def_33,type,
prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_34,type,
prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_35,type,
prefer270414042_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_36,type,
prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_37,type,
prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_38,type,
prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_39,type,
prefer1203288218_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_40,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_41,type,
prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_42,type,
prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_43,type,
prefer1671525929_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a ).
thf(func_def_44,type,
prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_45,type,
prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_46,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_47,type,
prefer1710114912_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_48,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_49,type,
prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_50,type,
prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_51,type,
prefer484977842_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_52,type,
prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_53,type,
prefer1064779160od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_54,type,
prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_55,type,
produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a ).
thf(func_def_56,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_57,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(func_def_58,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_59,type,
produc1926412547_set_a: produc1691597095_set_a > set_a ).
thf(func_def_60,type,
product_fst_a_a: product_prod_a_a > a ).
thf(func_def_61,type,
produc918972485_set_a: produc1691597095_set_a > set_a ).
thf(func_def_62,type,
product_snd_a_a: product_prod_a_a > a ).
thf(func_def_63,type,
product_swap_a_a: product_prod_a_a > product_prod_a_a ).
thf(func_def_64,type,
domain_a_a: set_Product_prod_a_a > set_a ).
thf(func_def_65,type,
id_on_1182990020od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_66,type,
id_on_a: set_a > set_Product_prod_a_a ).
thf(func_def_67,type,
range_a_a: set_Product_prod_a_a > set_a ).
thf(func_def_68,type,
refl_o198693862od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_69,type,
refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_70,type,
refl_on_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_71,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_72,type,
total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_73,type,
total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_74,type,
total_on_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_75,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_76,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_77,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(func_def_78,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_79,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_80,type,
insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_81,type,
insert_a: a > set_a > set_a ).
thf(func_def_82,type,
is_empty_a: set_a > $o ).
thf(func_def_83,type,
is_sin877026331od_a_a: set_Product_prod_a_a > $o ).
thf(func_def_84,type,
is_singleton_a: set_a > $o ).
thf(func_def_85,type,
the_elem_a: set_a > a ).
thf(func_def_86,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_87,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_88,type,
member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o ).
thf(func_def_89,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_90,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_91,type,
member_a: a > set_a > $o ).
thf(func_def_92,type,
carrier: set_a ).
thf(func_def_93,type,
relation: set_Product_prod_a_a ).
thf(func_def_94,type,
y: a ).
thf(func_def_95,type,
z: a ).
thf(f1049,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1041,f1047,f1048]) ).
thf(f1048,plain,
spl0_2,
inference(avatar_split_clause,[status(thm)],[f1044,f1038]) ).
thf(f1038,definition,
( spl0_2
<=> ( ( member_a @ z @ carrier )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition]) ).
thf(f1044,plain,
( ( member_a @ z @ carrier )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1043]) ).
thf(f1043,plain,
( ( $true != $true )
| ( ( member_a @ z @ carrier )
= $true ) ),
inference(superposition,[status(thm)],[f1029,f1018]) ).
thf(f1018,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation )
= $true ),
inference(cnf_transformation,[status(thm)],[f826]) ).
thf(f826,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation )
= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation )
= $true ) ),
inference(fool_elimination,[status(thm)],[f825]) ).
thf(f825,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation ) ),
file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',fact_0_assms) ).
thf(f1029,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
!= $true )
| ( ( member_a @ X1 @ carrier )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1010]) ).
thf(f1010,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
!= $true )
| ( ( member_a @ X1 @ carrier )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f832]) ).
thf(f832,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true )
=> ( ( member_a @ X1 @ carrier )
= $true ) ),
inference(fool_elimination,[status(thm)],[f831]) ).
thf(f831,plain,
! [X0: a,X1: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
=> ( member_a @ X1 @ carrier ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,axiom,
! [X3: a,X2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ X3 ) @ relation )
=> ( member_a @ X2 @ carrier ) ),
file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',fact_8_not__outside) ).
thf(f1047,plain,
spl0_1,
inference(avatar_contradiction_clause,[status(thm)],[f1046]) ).
thf(f1046,plain,
( $false
| spl0_1 ),
inference(subsumption_resolution,[status(thm)],[f1045,f1036]) ).
thf(f1036,plain,
( ( ( member_a @ y @ carrier )
!= $true )
| spl0_1 ),
inference(avatar_component_clause,[status(thm)],[f1034]) ).
thf(f1034,definition,
( spl0_1
<=> ( ( member_a @ y @ carrier )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f1045,plain,
( ( member_a @ y @ carrier )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1042]) ).
thf(f1042,plain,
( ( $true != $true )
| ( ( member_a @ y @ carrier )
= $true ) ),
inference(superposition,[status(thm)],[f1029,f1017]) ).
thf(f1017,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation )
= $true ),
inference(cnf_transformation,[status(thm)],[f826]) ).
thf(f1041,plain,
( ~ spl0_1
| ~ spl0_2 ),
inference(avatar_split_clause,[status(thm)],[f1015,f1038,f1034]) ).
thf(f1015,plain,
( ( ( member_a @ z @ carrier )
!= $true )
| ( ( member_a @ y @ carrier )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1005]) ).
thf(f1005,plain,
( ( ( member_a @ y @ carrier )
!= $true )
| ( ( member_a @ z @ carrier )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f824]) ).
thf(f824,plain,
~ ( ( ( member_a @ y @ carrier )
= $true )
& ( ( member_a @ z @ carrier )
= $true ) ),
inference(fool_elimination,[status(thm)],[f823]) ).
thf(f823,plain,
~ ( ( member_a @ y @ carrier )
& ( member_a @ z @ carrier ) ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
~ ( ( member_a @ y @ carrier )
& ( member_a @ z @ carrier ) ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
( ( member_a @ y @ carrier )
& ( member_a @ z @ carrier ) ),
file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP153^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.06 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.24 % Computer : n024.cluster.edu
% 0.07/0.24 % Model : x86_64 x86_64
% 0.07/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.24 % Memory : 8042.1875MB
% 0.07/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.24 % CPULimit : 300
% 0.07/0.24 % WCLimit : 300
% 0.07/0.24 % DateTime : Tue Feb 24 03:27:55 EST 2026
% 0.07/0.24 % CPUTime :
% 0.07/0.24 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : ITP153^1 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.31 % Domain : Interactive Theorem Proving
% 0.07/0.31 % Problem : Sledgehammer Preferences problem prob_180__6250282_1
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.31 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Des21]
% 0.07/0.31 % Names : Preferences/prob_180__6250282_1 [Des21]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.07/0.31 % Syntax : Number of formulae : 452 ( 134 unt; 96 typ; 0 def)
% 0.07/0.31 % Number of atoms : 1138 ( 279 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 10 ( 3 avg)
% 0.07/0.31 % Number of connectives : 3567 ( 147 ~; 29 |; 123 &;2778 @)
% 0.07/0.31 % ( 0 <=>; 490 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 16 ( 7 avg)
% 0.07/0.31 % Number of types : 12 ( 11 usr)
% 0.07/0.31 % Number of type conns : 183 ( 183 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 86 ( 85 usr; 9 con; 0-3 aty)
% 0.07/0.31 % Number of variables : 1027 ( 58 ^; 951 !; 18 ?;1027 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : This file was generated by Sledgehammer 2021-02-23 15:44:16.629
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % Could-be-implicit typings (11)
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% 0.07/0.31 total_on_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Relation_Ototal__on_001tf__a,type,
% 0.07/0.31 total_on_a: set_a > set_Product_prod_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Relation_Otrans_001tf__a,type,
% 0.07/0.31 trans_a: set_Product_prod_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.07/0.31 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.07/0.31 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.07/0.31 collect_a: ( a > $o ) > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.07/0.31 insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oinsert_001tf__a,type,
% 0.07/0.31 insert_a: a > set_a > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 0.07/0.31 is_empty_a: set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.07/0.31 is_sin877026331od_a_a: set_Product_prod_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 0.07/0.31 is_singleton_a: set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 0.07/0.31 the_elem_a: set_a > a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.07/0.31 member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.07/0.31 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.07/0.31 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 member_set_a: set_a > set_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001tf__a,type,
% 0.07/0.31 member_a: a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_carrier,type,
% 0.07/0.31 carrier: set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_relation,type,
% 0.07/0.31 relation: set_Product_prod_a_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_y,type,
% 0.07/0.31 y: a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_z,type,
% 0.07/0.31 z: a ).
% 0.07/0.31
% 0.07/0.31 % Relevant facts (355)
% 0.07/0.31 thf(fact_0_assms,axiom,
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % assms
% 0.07/0.31 thf(fact_1_rational__preference__axioms,axiom,
% 0.07/0.31 prefer719835986ence_a @ carrier @ relation ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_axioms
% 0.07/0.31 thf(fact_2_preference__axioms,axiom,
% 0.07/0.31 prefer1292084992ence_a @ carrier @ relation ).
% 0.07/0.31
% 0.07/0.31 % preference_axioms
% 0.07/0.31 thf(fact_3_trans__refl,axiom,
% 0.07/0.31 order_preorder_on_a @ carrier @ relation ).
% 0.07/0.31
% 0.07/0.31 % trans_refl
% 0.07/0.31 thf(fact_4_total,axiom,
% 0.07/0.31 total_on_a @ carrier @ relation ).
% 0.07/0.31
% 0.07/0.31 % total
% 0.07/0.31 thf(fact_5_reflexivity,axiom,
% 0.07/0.31 refl_on_a @ carrier @ relation ).
% 0.07/0.31
% 0.07/0.31 % reflexivity
% 0.07/0.31 thf(fact_6_compl,axiom,
% 0.07/0.31 ! [X: a] :
% 0.07/0.31 ( ( member_a @ X @ carrier )
% 0.07/0.31 => ! [Xa: a] :
% 0.07/0.31 ( ( member_a @ Xa @ carrier )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ relation )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % compl
% 0.07/0.31 thf(fact_7_completeD,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member_a @ X2 @ carrier )
% 0.07/0.31 => ( ( member_a @ Y @ carrier )
% 0.07/0.31 => ( ( X2 != Y )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % completeD
% 0.07/0.31 thf(fact_8_not__outside,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 => ( member_a @ X2 @ carrier ) ) ).
% 0.07/0.31
% 0.07/0.31 % not_outside
% 0.07/0.31 thf(fact_9_strict__is__neg__transitive,axiom,
% 0.07/0.31 ! [X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( ( member_a @ X2 @ carrier )
% 0.07/0.31 & ( member_a @ Y @ carrier )
% 0.07/0.31 & ( member_a @ Z @ carrier ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) )
% 0.07/0.31 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % strict_is_neg_transitive
% 0.07/0.31 thf(fact_10_weak__is__transitive,axiom,
% 0.07/0.31 ! [X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( ( member_a @ X2 @ carrier )
% 0.07/0.31 & ( member_a @ Y @ carrier )
% 0.07/0.31 & ( member_a @ Z @ carrier ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % weak_is_transitive
% 0.07/0.31 thf(fact_11_strict__not__refl__weak,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( ( member_a @ X2 @ carrier )
% 0.07/0.31 & ( member_a @ Y @ carrier ) )
% 0.07/0.31 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 0.07/0.31 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % strict_not_refl_weak
% 0.07/0.31 thf(fact_12_indiff__trans,axiom,
% 0.07/0.31 ! [X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % indiff_trans
% 0.07/0.31 thf(fact_13_strict__trans,axiom,
% 0.07/0.31 ! [X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % strict_trans
% 0.07/0.31 thf(fact_14_pref__in__at__least__as,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 => ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % pref_in_at_least_as
% 0.07/0.31 thf(fact_15_rational__preference_Oaxioms_I1_J,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.axioms(1)
% 0.07/0.31 thf(fact_16_rational__preference_Oaxioms_I1_J,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.axioms(1)
% 0.07/0.31 thf(fact_17_preference__def,axiom,
% 0.07/0.31 ( prefer1409946857od_a_a
% 0.07/0.31 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 0.07/0.31 ( ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 0.07/0.31 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member2057358096od_a_a @ X3 @ Carrier2 ) )
% 0.07/0.31 & ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 0.07/0.31 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member2057358096od_a_a @ Y2 @ Carrier2 ) )
% 0.07/0.31 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference_def
% 0.07/0.31 thf(fact_18_preference__def,axiom,
% 0.07/0.31 ( prefer1710114912_set_a
% 0.07/0.31 = ( ^ [Carrier2: set_set_a,Relation2: set_Pr2045688071_set_a] :
% 0.07/0.31 ( ! [X3: set_a,Y2: set_a] :
% 0.07/0.31 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member_set_a @ X3 @ Carrier2 ) )
% 0.07/0.31 & ! [X3: set_a,Y2: set_a] :
% 0.07/0.31 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member_set_a @ Y2 @ Carrier2 ) )
% 0.07/0.31 & ( order_816746613_set_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference_def
% 0.07/0.31 thf(fact_19_preference__def,axiom,
% 0.07/0.31 ( prefer1338892073od_a_a
% 0.07/0.31 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 0.07/0.31 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 0.07/0.31 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference_def
% 0.07/0.31 thf(fact_20_preference__def,axiom,
% 0.07/0.31 ( prefer1292084992ence_a
% 0.07/0.31 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X3: a,Y2: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member_a @ X3 @ Carrier2 ) )
% 0.07/0.31 & ! [X3: a,Y2: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.07/0.31 => ( member_a @ Y2 @ Carrier2 ) )
% 0.07/0.31 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference_def
% 0.07/0.31 thf(fact_21_preference_Ointro,axiom,
% 0.07/0.31 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.07/0.31 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ X4 @ Carrier ) )
% 0.07/0.31 => ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.07/0.31 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 0.07/0.31 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.intro
% 0.07/0.31 thf(fact_22_preference_Ointro,axiom,
% 0.07/0.31 ! [Relation: set_Pr2045688071_set_a,Carrier: set_set_a] :
% 0.07/0.31 ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member_set_a @ X4 @ Carrier ) )
% 0.07/0.31 => ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member_set_a @ Y3 @ Carrier ) )
% 0.07/0.31 => ( ( order_816746613_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1710114912_set_a @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.intro
% 0.07/0.31 thf(fact_23_preference_Ointro,axiom,
% 0.07/0.31 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 0.07/0.31 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 0.07/0.31 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.intro
% 0.07/0.31 thf(fact_24_preference_Ointro,axiom,
% 0.07/0.31 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 0.07/0.31 ( ! [X4: a,Y3: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member_a @ X4 @ Carrier ) )
% 0.07/0.31 => ( ! [X4: a,Y3: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.07/0.31 => ( member_a @ Y3 @ Carrier ) )
% 0.07/0.31 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.intro
% 0.07/0.31 thf(fact_25_preference_Otrans__refl,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.trans_refl
% 0.07/0.31 thf(fact_26_preference_Otrans__refl,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.trans_refl
% 0.07/0.31 thf(fact_27_preference_Onot__outside,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ X2 @ Carrier ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.not_outside
% 0.07/0.31 thf(fact_28_preference_Onot__outside,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_set_a @ X2 @ Carrier ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.not_outside
% 0.07/0.31 thf(fact_29_preference_Onot__outside,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ X2 @ Carrier ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.not_outside
% 0.07/0.31 thf(fact_30_preference_Onot__outside,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_a @ X2 @ Carrier ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.not_outside
% 0.07/0.31 thf(fact_31_preference_Oreflexivity,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.reflexivity
% 0.07/0.31 thf(fact_32_preference_Oreflexivity,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.reflexivity
% 0.07/0.31 thf(fact_33_preference_Oindiff__trans,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 0.07/0.31 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.indiff_trans
% 0.07/0.31 thf(fact_34_preference_Oindiff__trans,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.indiff_trans
% 0.07/0.31 thf(fact_35_preference_Oindiff__trans,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.indiff_trans
% 0.07/0.31 thf(fact_36_rational__preference_Ocompl,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ! [X: set_a] :
% 0.07/0.31 ( ( member_set_a @ X @ Carrier )
% 0.07/0.31 => ! [Xa: set_a] :
% 0.07/0.31 ( ( member_set_a @ Xa @ Carrier )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Xa ) @ Relation )
% 0.07/0.31 | ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.compl
% 0.07/0.31 thf(fact_37_rational__preference_Ocompl,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ! [X: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X @ Carrier )
% 0.07/0.31 => ! [Xa: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ Xa @ Carrier )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Xa ) @ Relation )
% 0.07/0.31 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.compl
% 0.07/0.31 thf(fact_38_rational__preference_Ocompl,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ! [X: a] :
% 0.07/0.31 ( ( member_a @ X @ Carrier )
% 0.07/0.31 => ! [Xa: a] :
% 0.07/0.31 ( ( member_a @ Xa @ Carrier )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ Relation )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.compl
% 0.07/0.31 thf(fact_39_rational__preference_Ototal,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.total
% 0.07/0.31 thf(fact_40_rational__preference_Ototal,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( total_on_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.total
% 0.07/0.31 thf(fact_41_rational__preference_OcompleteD,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ( X2 != Y )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.completeD
% 0.07/0.31 thf(fact_42_rational__preference_OcompleteD,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_set_a @ Y @ Carrier )
% 0.07/0.31 => ( ( X2 != Y )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.completeD
% 0.07/0.31 thf(fact_43_rational__preference_OcompleteD,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ( X2 != Y )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.completeD
% 0.07/0.31 thf(fact_44_rational__preference_OcompleteD,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_a @ Y @ Carrier )
% 0.07/0.31 => ( ( X2 != Y )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.completeD
% 0.07/0.31 thf(fact_45_rational__preference_Ostrict__trans,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_trans
% 0.07/0.31 thf(fact_46_rational__preference_Ostrict__trans,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_trans
% 0.07/0.31 thf(fact_47_rational__preference_Ostrict__trans,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_trans
% 0.07/0.31 thf(fact_48_rational__preference_Oweak__is__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member2057358096od_a_a @ Y @ Carrier )
% 0.07/0.31 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.weak_is_transitive
% 0.07/0.31 thf(fact_49_rational__preference_Oweak__is__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_set_a @ Y @ Carrier )
% 0.07/0.31 & ( member_set_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 0.07/0.31 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.weak_is_transitive
% 0.07/0.31 thf(fact_50_rational__preference_Oweak__is__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member449909584od_a_a @ Y @ Carrier )
% 0.07/0.31 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.weak_is_transitive
% 0.07/0.31 thf(fact_51_rational__preference_Oweak__is__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_a @ Y @ Carrier )
% 0.07/0.31 & ( member_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.weak_is_transitive
% 0.07/0.31 thf(fact_52_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 0.07/0.31 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_not_refl_weak
% 0.07/0.31 thf(fact_53_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_set_a @ Y @ Carrier ) )
% 0.07/0.31 => ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_not_refl_weak
% 0.07/0.31 thf(fact_54_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member449909584od_a_a @ Y @ Carrier ) )
% 0.07/0.31 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_not_refl_weak
% 0.07/0.31 thf(fact_55_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_a @ Y @ Carrier ) )
% 0.07/0.31 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_not_refl_weak
% 0.07/0.31 thf(fact_56_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member2057358096od_a_a @ Y @ Carrier )
% 0.07/0.31 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X2 ) @ Relation ) )
% 0.07/0.31 | ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_is_neg_transitive
% 0.07/0.31 thf(fact_57_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_set_a @ Y @ Carrier )
% 0.07/0.31 & ( member_set_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) )
% 0.07/0.31 | ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_is_neg_transitive
% 0.07/0.31 thf(fact_58_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 & ( member449909584od_a_a @ Y @ Carrier )
% 0.07/0.31 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) )
% 0.07/0.31 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_is_neg_transitive
% 0.07/0.31 thf(fact_59_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 & ( member_a @ Y @ Carrier )
% 0.07/0.31 & ( member_a @ Z @ Carrier ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 0.07/0.31 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) )
% 0.07/0.31 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.strict_is_neg_transitive
% 0.07/0.31 thf(fact_60_worse__in__no__better,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % worse_in_no_better
% 0.07/0.31 thf(fact_61_as__good__as__sameIff,axiom,
% 0.07/0.31 ! [Z: a,Y: a] :
% 0.07/0.31 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 0.07/0.31 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % as_good_as_sameIff
% 0.07/0.31 thf(fact_62_prod_Oinject,axiom,
% 0.07/0.31 ! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y22: product_prod_a_a] :
% 0.07/0.31 ( ( ( produc1474507607od_a_a @ X1 @ X22 )
% 0.07/0.31 = ( produc1474507607od_a_a @ Y1 @ Y22 ) )
% 0.07/0.31 = ( ( X1 = Y1 )
% 0.07/0.31 & ( X22 = Y22 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod.inject
% 0.07/0.31 thf(fact_63_prod_Oinject,axiom,
% 0.07/0.31 ! [X1: set_a,X22: set_a,Y1: set_a,Y22: set_a] :
% 0.07/0.31 ( ( ( produc1928581911_set_a @ X1 @ X22 )
% 0.07/0.31 = ( produc1928581911_set_a @ Y1 @ Y22 ) )
% 0.07/0.31 = ( ( X1 = Y1 )
% 0.07/0.31 & ( X22 = Y22 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod.inject
% 0.07/0.31 thf(fact_64_prod_Oinject,axiom,
% 0.07/0.31 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 0.07/0.31 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 0.07/0.31 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 0.07/0.31 = ( ( X1 = Y1 )
% 0.07/0.31 & ( X22 = Y22 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod.inject
% 0.07/0.31 thf(fact_65_old_Oprod_Oinject,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 0.07/0.31 ( ( ( produc1474507607od_a_a @ A @ B )
% 0.07/0.31 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 0.07/0.31 = ( ( A = A2 )
% 0.07/0.31 & ( B = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inject
% 0.07/0.31 thf(fact_66_old_Oprod_Oinject,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 0.07/0.31 ( ( ( produc1928581911_set_a @ A @ B )
% 0.07/0.31 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 0.07/0.31 = ( ( A = A2 )
% 0.07/0.31 & ( B = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inject
% 0.07/0.31 thf(fact_67_old_Oprod_Oinject,axiom,
% 0.07/0.31 ! [A: a,B: a,A2: a,B2: a] :
% 0.07/0.31 ( ( ( product_Pair_a_a @ A @ B )
% 0.07/0.31 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.07/0.31 = ( ( A = A2 )
% 0.07/0.31 & ( B = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inject
% 0.07/0.31 thf(fact_68_rational__preference_Oas__good__as__sameIff,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,Z: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ Z @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.as_good_as_sameIff
% 0.07/0.31 thf(fact_69_rational__preference_Oas__good__as__sameIff,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_set_a @ Z @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.as_good_as_sameIff
% 0.07/0.31 thf(fact_70_rational__preference_Oas__good__as__sameIff,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.as_good_as_sameIff
% 0.07/0.31 thf(fact_71_rational__preference_Oas__good__as__sameIff,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.as_good_as_sameIff
% 0.07/0.31 thf(fact_72_total__onI,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a] :
% 0.07/0.31 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member2057358096od_a_a @ Y3 @ A3 )
% 0.07/0.31 => ( ( X4 != Y3 )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.07/0.31 => ( total_1340260971od_a_a @ A3 @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_onI
% 0.07/0.31 thf(fact_73_total__onI,axiom,
% 0.07/0.31 ! [A3: set_set_a,R: set_Pr2045688071_set_a] :
% 0.07/0.31 ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( member_set_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member_set_a @ Y3 @ A3 )
% 0.07/0.31 => ( ( X4 != Y3 )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.07/0.31 => ( total_on_set_a @ A3 @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_onI
% 0.07/0.31 thf(fact_74_total__onI,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member449909584od_a_a @ Y3 @ A3 )
% 0.07/0.31 => ( ( X4 != Y3 )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.07/0.31 => ( total_1490170027od_a_a @ A3 @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_onI
% 0.07/0.31 thf(fact_75_total__onI,axiom,
% 0.07/0.31 ! [A3: set_a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: a,Y3: a] :
% 0.07/0.31 ( ( member_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member_a @ Y3 @ A3 )
% 0.07/0.31 => ( ( X4 != Y3 )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.07/0.31 => ( total_on_a @ A3 @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_onI
% 0.07/0.31 thf(fact_76_total__on__def,axiom,
% 0.07/0.31 ( total_on_set_a
% 0.07/0.31 = ( ^ [A4: set_set_a,R2: set_Pr2045688071_set_a] :
% 0.07/0.31 ! [X3: set_a] :
% 0.07/0.31 ( ( member_set_a @ X3 @ A4 )
% 0.07/0.31 => ! [Y2: set_a] :
% 0.07/0.31 ( ( member_set_a @ Y2 @ A4 )
% 0.07/0.31 => ( ( X3 != Y2 )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ R2 )
% 0.07/0.31 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_def
% 0.07/0.31 thf(fact_77_total__on__def,axiom,
% 0.07/0.31 ( total_1490170027od_a_a
% 0.07/0.31 = ( ^ [A4: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 0.07/0.31 ! [X3: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.07/0.31 => ! [Y2: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ Y2 @ A4 )
% 0.07/0.31 => ( ( X3 != Y2 )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ R2 )
% 0.07/0.31 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_def
% 0.07/0.31 thf(fact_78_total__on__def,axiom,
% 0.07/0.31 ( total_on_a
% 0.07/0.31 = ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
% 0.07/0.31 ! [X3: a] :
% 0.07/0.31 ( ( member_a @ X3 @ A4 )
% 0.07/0.31 => ! [Y2: a] :
% 0.07/0.31 ( ( member_a @ Y2 @ A4 )
% 0.07/0.31 => ( ( X3 != Y2 )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 0.07/0.31 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_def
% 0.07/0.31 thf(fact_79_refl__onD,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
% 0.07/0.31 ( ( refl_o198693862od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member2057358096od_a_a @ A @ A3 )
% 0.07/0.31 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD
% 0.07/0.31 thf(fact_80_refl__onD,axiom,
% 0.07/0.31 ! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a] :
% 0.07/0.31 ( ( refl_on_set_a @ A3 @ R )
% 0.07/0.31 => ( ( member_set_a @ A @ A3 )
% 0.07/0.31 => ( member593450832_set_a @ ( produc1928581911_set_a @ A @ A ) @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD
% 0.07/0.31 thf(fact_81_refl__onD,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 0.07/0.31 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD
% 0.07/0.31 thf(fact_82_refl__onD,axiom,
% 0.07/0.31 ! [A3: set_a,R: set_Product_prod_a_a,A: a] :
% 0.07/0.31 ( ( refl_on_a @ A3 @ R )
% 0.07/0.31 => ( ( member_a @ A @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD
% 0.07/0.31 thf(fact_83_refl__onD1,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( refl_o198693862od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member2057358096od_a_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD1
% 0.07/0.31 thf(fact_84_refl__onD1,axiom,
% 0.07/0.31 ! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( refl_on_set_a @ A3 @ R )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member_set_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD1
% 0.07/0.31 thf(fact_85_refl__onD1,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD1
% 0.07/0.31 thf(fact_86_refl__onD1,axiom,
% 0.07/0.31 ! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( refl_on_a @ A3 @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD1
% 0.07/0.31 thf(fact_87_refl__onD2,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( refl_o198693862od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member2057358096od_a_a @ Y @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD2
% 0.07/0.31 thf(fact_88_refl__onD2,axiom,
% 0.07/0.31 ! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( refl_on_set_a @ A3 @ R )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member_set_a @ Y @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD2
% 0.07/0.31 thf(fact_89_refl__onD2,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ Y @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD2
% 0.07/0.31 thf(fact_90_refl__onD2,axiom,
% 0.07/0.31 ! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( refl_on_a @ A3 @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( member_a @ Y @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_onD2
% 0.07/0.31 thf(fact_91_refl__on__domain,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
% 0.07/0.31 ( ( refl_o198693862od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R )
% 0.07/0.31 => ( ( member2057358096od_a_a @ A @ A3 )
% 0.07/0.31 & ( member2057358096od_a_a @ B @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_domain
% 0.07/0.31 thf(fact_92_refl__on__domain,axiom,
% 0.07/0.31 ! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a,B: set_a] :
% 0.07/0.31 ( ( refl_on_set_a @ A3 @ R )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ A @ B ) @ R )
% 0.07/0.31 => ( ( member_set_a @ A @ A3 )
% 0.07/0.31 & ( member_set_a @ B @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_domain
% 0.07/0.31 thf(fact_93_refl__on__domain,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 0.07/0.31 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 & ( member449909584od_a_a @ B @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_domain
% 0.07/0.31 thf(fact_94_refl__on__domain,axiom,
% 0.07/0.31 ! [A3: set_a,R: set_Product_prod_a_a,A: a,B: a] :
% 0.07/0.31 ( ( refl_on_a @ A3 @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 0.07/0.31 => ( ( member_a @ A @ A3 )
% 0.07/0.31 & ( member_a @ B @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_domain
% 0.07/0.31 thf(fact_95_at__least__as__goodD,axiom,
% 0.07/0.31 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_least_as_goodD
% 0.07/0.31 thf(fact_96_at__least__as__goodD,axiom,
% 0.07/0.31 ! [Z: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
% 0.07/0.31 ( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_least_as_goodD
% 0.07/0.31 thf(fact_97_at__least__as__goodD,axiom,
% 0.07/0.31 ! [Z: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_least_as_goodD
% 0.07/0.31 thf(fact_98_at__least__as__goodD,axiom,
% 0.07/0.31 ! [Z: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_least_as_goodD
% 0.07/0.31 thf(fact_99_at__lst__asgd__ge,axiom,
% 0.07/0.31 ! [X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_lst_asgd_ge
% 0.07/0.31 thf(fact_100_at__lst__asgd__ge,axiom,
% 0.07/0.31 ! [X2: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
% 0.07/0.31 ( ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_lst_asgd_ge
% 0.07/0.31 thf(fact_101_at__lst__asgd__ge,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_lst_asgd_ge
% 0.07/0.31 thf(fact_102_at__lst__asgd__ge,axiom,
% 0.07/0.31 ! [X2: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ X2 @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Pr ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_lst_asgd_ge
% 0.07/0.31 thf(fact_103_old_Oprod_Oinducts,axiom,
% 0.07/0.31 ! [P: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 0.07/0.31 ( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ Prod ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inducts
% 0.07/0.31 thf(fact_104_old_Oprod_Oinducts,axiom,
% 0.07/0.31 ! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
% 0.07/0.31 ( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ Prod ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inducts
% 0.07/0.31 thf(fact_105_old_Oprod_Oinducts,axiom,
% 0.07/0.31 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.07/0.31 ( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ Prod ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.inducts
% 0.07/0.31 thf(fact_106_old_Oprod_Oexhaust,axiom,
% 0.07/0.31 ! [Y: produc1572603623od_a_a] :
% 0.07/0.31 ~ ! [A5: product_prod_a_a,B4: product_prod_a_a] :
% 0.07/0.31 ( Y
% 0.07/0.31 != ( produc1474507607od_a_a @ A5 @ B4 ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.exhaust
% 0.07/0.31 thf(fact_107_old_Oprod_Oexhaust,axiom,
% 0.07/0.31 ! [Y: produc1691597095_set_a] :
% 0.07/0.31 ~ ! [A5: set_a,B4: set_a] :
% 0.07/0.31 ( Y
% 0.07/0.31 != ( produc1928581911_set_a @ A5 @ B4 ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.exhaust
% 0.07/0.31 thf(fact_108_old_Oprod_Oexhaust,axiom,
% 0.07/0.31 ! [Y: product_prod_a_a] :
% 0.07/0.31 ~ ! [A5: a,B4: a] :
% 0.07/0.31 ( Y
% 0.07/0.31 != ( product_Pair_a_a @ A5 @ B4 ) ) ).
% 0.07/0.31
% 0.07/0.31 % old.prod.exhaust
% 0.07/0.31 thf(fact_109_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
% 0.07/0.31 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P ) )
% 0.07/0.31 = ( P @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_110_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A: a,P: a > $o] :
% 0.07/0.31 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.07/0.31 = ( P @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_111_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.07/0.31 = ( P @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_112_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A3: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( collec1635618130od_a_a
% 0.07/0.31 @ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A3 ) )
% 0.07/0.31 = A3 ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_113_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( collect_a
% 0.07/0.31 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 0.07/0.31 = A3 ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_114_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( collec645855634od_a_a
% 0.07/0.31 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 0.07/0.31 = A3 ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_115_Collect__cong,axiom,
% 0.07/0.31 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.07/0.31 ( ! [X4: product_prod_a_a] :
% 0.07/0.31 ( ( P @ X4 )
% 0.07/0.31 = ( Q @ X4 ) )
% 0.07/0.31 => ( ( collec645855634od_a_a @ P )
% 0.07/0.31 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_116_Collect__cong,axiom,
% 0.07/0.31 ! [P: a > $o,Q: a > $o] :
% 0.07/0.31 ( ! [X4: a] :
% 0.07/0.31 ( ( P @ X4 )
% 0.07/0.31 = ( Q @ X4 ) )
% 0.07/0.31 => ( ( collect_a @ P )
% 0.07/0.31 = ( collect_a @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_117_prod__induct3,axiom,
% 0.07/0.31 ! [P: produc1572603623od_a_a > $o,X2: produc1572603623od_a_a] :
% 0.07/0.31 ( ! [A5: product_prod_a_a,B4: a,C: a] : ( P @ ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) )
% 0.07/0.31 => ( P @ X2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod_induct3
% 0.07/0.31 thf(fact_118_prod__cases3,axiom,
% 0.07/0.31 ! [Y: produc1572603623od_a_a] :
% 0.07/0.31 ~ ! [A5: product_prod_a_a,B4: a,C: a] :
% 0.07/0.31 ( Y
% 0.07/0.31 != ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod_cases3
% 0.07/0.31 thf(fact_119_Pair__inject,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 0.07/0.31 ( ( ( produc1474507607od_a_a @ A @ B )
% 0.07/0.31 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 0.07/0.31 => ~ ( ( A = A2 )
% 0.07/0.31 => ( B != B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Pair_inject
% 0.07/0.31 thf(fact_120_Pair__inject,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 0.07/0.31 ( ( ( produc1928581911_set_a @ A @ B )
% 0.07/0.31 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 0.07/0.31 => ~ ( ( A = A2 )
% 0.07/0.31 => ( B != B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Pair_inject
% 0.07/0.31 thf(fact_121_Pair__inject,axiom,
% 0.07/0.31 ! [A: a,B: a,A2: a,B2: a] :
% 0.07/0.31 ( ( ( product_Pair_a_a @ A @ B )
% 0.07/0.31 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.07/0.31 => ~ ( ( A = A2 )
% 0.07/0.31 => ( B != B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Pair_inject
% 0.07/0.31 thf(fact_122_prod__cases,axiom,
% 0.07/0.31 ! [P: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a] :
% 0.07/0.31 ( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ P2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod_cases
% 0.07/0.31 thf(fact_123_prod__cases,axiom,
% 0.07/0.31 ! [P: produc1691597095_set_a > $o,P2: produc1691597095_set_a] :
% 0.07/0.31 ( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ P2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod_cases
% 0.07/0.31 thf(fact_124_prod__cases,axiom,
% 0.07/0.31 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.07/0.31 ( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
% 0.07/0.31 => ( P @ P2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod_cases
% 0.07/0.31 thf(fact_125_surj__pair,axiom,
% 0.07/0.31 ! [P2: produc1572603623od_a_a] :
% 0.07/0.31 ? [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.07/0.31 ( P2
% 0.07/0.31 = ( produc1474507607od_a_a @ X4 @ Y3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % surj_pair
% 0.07/0.31 thf(fact_126_surj__pair,axiom,
% 0.07/0.31 ! [P2: produc1691597095_set_a] :
% 0.07/0.31 ? [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( P2
% 0.07/0.31 = ( produc1928581911_set_a @ X4 @ Y3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % surj_pair
% 0.07/0.31 thf(fact_127_surj__pair,axiom,
% 0.07/0.31 ! [P2: product_prod_a_a] :
% 0.07/0.31 ? [X4: a,Y3: a] :
% 0.07/0.31 ( P2
% 0.07/0.31 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % surj_pair
% 0.07/0.31 thf(fact_128_rational__preference_Oworse__in__no__better,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.worse_in_no_better
% 0.07/0.31 thf(fact_129_rational__preference_Oworse__in__no__better,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.worse_in_no_better
% 0.07/0.31 thf(fact_130_rational__preference_Oworse__in__no__better,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.worse_in_no_better
% 0.07/0.31 thf(fact_131_rational__preference_Oworse__in__no__better,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.worse_in_no_better
% 0.07/0.31 thf(fact_132_rational__preference_Opref__in__at__least__as,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.pref_in_at_least_as
% 0.07/0.31 thf(fact_133_rational__preference_Opref__in__at__least__as,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.pref_in_at_least_as
% 0.07/0.31 thf(fact_134_rational__preference_Opref__in__at__least__as,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.pref_in_at_least_as
% 0.07/0.31 thf(fact_135_rational__preference_Opref__in__at__least__as,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.pref_in_at_least_as
% 0.07/0.31 thf(fact_136_at__lst__asgd__not__ge,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( carrier != bot_bot_set_a )
% 0.07/0.31 => ( ( member_a @ X2 @ carrier )
% 0.07/0.31 => ( ( member_a @ Y @ carrier )
% 0.07/0.31 => ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 0.07/0.31 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % at_lst_asgd_not_ge
% 0.07/0.31 thf(fact_137_no__better__subset__pref,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % no_better_subset_pref
% 0.07/0.31 thf(fact_138_no__better__thansubset__rel,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( member_a @ X2 @ carrier )
% 0.07/0.31 => ( ( member_a @ Y @ carrier )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % no_better_thansubset_rel
% 0.07/0.31 thf(fact_139_no__better__than__nonepty,axiom,
% 0.07/0.31 ! [X2: a] :
% 0.07/0.31 ( ( carrier != bot_bot_set_a )
% 0.07/0.31 => ( ( member_a @ X2 @ carrier )
% 0.07/0.31 => ( ( prefer1676310729than_a @ X2 @ carrier @ relation )
% 0.07/0.31 != bot_bot_set_a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % no_better_than_nonepty
% 0.07/0.31 thf(fact_140_nbt__nest,axiom,
% 0.07/0.31 ! [Y: a,X2: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
% 0.07/0.31 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nbt_nest
% 0.07/0.31 thf(fact_141_rational__preference_Ointro,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( prefer1064779160od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.intro
% 0.07/0.31 thf(fact_142_rational__preference_Ointro,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.intro
% 0.07/0.31 thf(fact_143_rational__preference__def,axiom,
% 0.07/0.31 ( prefer697821563od_a_a
% 0.07/0.31 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
% 0.07/0.31 & ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_def
% 0.07/0.31 thf(fact_144_rational__preference__def,axiom,
% 0.07/0.31 ( prefer719835986ence_a
% 0.07/0.31 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 0.07/0.31 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_def
% 0.07/0.31 thf(fact_145_refl__on__empty,axiom,
% 0.07/0.31 refl_o1298442278od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% 0.07/0.31
% 0.07/0.31 % refl_on_empty
% 0.07/0.31 thf(fact_146_refl__on__empty,axiom,
% 0.07/0.31 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.07/0.31
% 0.07/0.31 % refl_on_empty
% 0.07/0.31 thf(fact_147_preorder__on__empty,axiom,
% 0.07/0.31 order_359341630od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% 0.07/0.31
% 0.07/0.31 % preorder_on_empty
% 0.07/0.31 thf(fact_148_preorder__on__empty,axiom,
% 0.07/0.31 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.07/0.31
% 0.07/0.31 % preorder_on_empty
% 0.07/0.31 thf(fact_149_total__on__empty,axiom,
% 0.07/0.31 ! [R: set_Pr1948701895od_a_a] : ( total_1490170027od_a_a @ bot_bo2131659635od_a_a @ R ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_empty
% 0.07/0.31 thf(fact_150_total__on__empty,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_empty
% 0.07/0.31 thf(fact_151_rational__preference_Onbt__nest,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X2: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.nbt_nest
% 0.07/0.31 thf(fact_152_rational__preference_Onbt__nest,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X2: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.nbt_nest
% 0.07/0.31 thf(fact_153_rational__preference_Ono__better__than__nonepty,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bo1116205875od_a_a )
% 0.07/0.31 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( prefer926864306od_a_a @ X2 @ Carrier @ Relation )
% 0.07/0.31 != bot_bo1116205875od_a_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_than_nonepty
% 0.07/0.31 thf(fact_154_rational__preference_Ono__better__than__nonepty,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.07/0.31 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation )
% 0.07/0.31 != bot_bo2131659635od_a_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_than_nonepty
% 0.07/0.31 thf(fact_155_rational__preference_Ono__better__than__nonepty,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bot_set_a )
% 0.07/0.31 => ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( prefer1676310729than_a @ X2 @ Carrier @ Relation )
% 0.07/0.31 != bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_than_nonepty
% 0.07/0.31 thf(fact_156_rational__preference__axioms__def,axiom,
% 0.07/0.31 prefer1064779160od_a_a = total_1490170027od_a_a ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_axioms_def
% 0.07/0.31 thf(fact_157_rational__preference__axioms__def,axiom,
% 0.07/0.31 prefer580340847ioms_a = total_on_a ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_axioms_def
% 0.07/0.31 thf(fact_158_rational__preference__axioms_Ointro,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( total_1490170027od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_axioms.intro
% 0.07/0.31 thf(fact_159_rational__preference__axioms_Ointro,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( total_on_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference_axioms.intro
% 0.07/0.31 thf(fact_160_rational__preference_Oaxioms_I2_J,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.axioms(2)
% 0.07/0.31 thf(fact_161_rational__preference_Oaxioms_I2_J,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.axioms(2)
% 0.07/0.31 thf(fact_162_rational__preference_Ono__better__subset__pref,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_subset_pref
% 0.07/0.31 thf(fact_163_rational__preference_Ono__better__subset__pref,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_subset_pref
% 0.07/0.31 thf(fact_164_rational__preference_Ono__better__subset__pref,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_subset_pref
% 0.07/0.31 thf(fact_165_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_thansubset_rel
% 0.07/0.31 thf(fact_166_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_set_a @ Y @ Carrier )
% 0.07/0.31 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_thansubset_rel
% 0.07/0.31 thf(fact_167_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_thansubset_rel
% 0.07/0.31 thf(fact_168_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_a @ Y @ Carrier )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.no_better_thansubset_rel
% 0.07/0.31 thf(fact_169_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.07/0.31 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.07/0.31 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bo1116205875od_a_a )
% 0.07/0.31 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ~ ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 => ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.at_lst_asgd_not_ge
% 0.07/0.31 thf(fact_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.07/0.31 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bot_set_set_a )
% 0.07/0.31 => ( ( member_set_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_set_a @ Y @ Carrier )
% 0.07/0.31 => ( ~ ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 => ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.at_lst_asgd_not_ge
% 0.07/0.31 thf(fact_171_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.07/0.31 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 0.07/0.31 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.07/0.31 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.07/0.31 => ( ~ ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.at_lst_asgd_not_ge
% 0.07/0.31 thf(fact_172_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 0.07/0.31 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( ( Carrier != bot_bot_set_a )
% 0.07/0.31 => ( ( member_a @ X2 @ Carrier )
% 0.07/0.31 => ( ( member_a @ Y @ Carrier )
% 0.07/0.31 => ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 0.07/0.31 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rational_preference.at_lst_asgd_not_ge
% 0.07/0.31 thf(fact_173_subset__empty,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
% 0.07/0.31 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_empty
% 0.07/0.31 thf(fact_174_subset__empty,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 0.07/0.31 = ( A3 = bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_empty
% 0.07/0.31 thf(fact_175_empty__subsetI,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% 0.07/0.31
% 0.07/0.31 % empty_subsetI
% 0.07/0.31 thf(fact_176_empty__subsetI,axiom,
% 0.07/0.31 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 0.07/0.31
% 0.07/0.31 % empty_subsetI
% 0.07/0.31 thf(fact_177_Pair__le,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,C2: set_a,D: set_a] :
% 0.07/0.31 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C2 @ D ) )
% 0.07/0.31 = ( ( ord_less_eq_set_a @ A @ C2 )
% 0.07/0.31 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Pair_le
% 0.07/0.31 thf(fact_178_subsetI,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ X4 @ B3 ) )
% 0.07/0.31 => ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetI
% 0.07/0.31 thf(fact_179_subsetI,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ! [X4: a] :
% 0.07/0.31 ( ( member_a @ X4 @ A3 )
% 0.07/0.31 => ( member_a @ X4 @ B3 ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetI
% 0.07/0.31 thf(fact_180_subset__antisym,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 0.07/0.31 => ( A3 = B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_antisym
% 0.07/0.31 thf(fact_181_empty__iff,axiom,
% 0.07/0.31 ! [C2: product_prod_a_a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 0.07/0.31
% 0.07/0.31 % empty_iff
% 0.07/0.31 thf(fact_182_empty__iff,axiom,
% 0.07/0.31 ! [C2: a] :
% 0.07/0.31 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 0.07/0.31
% 0.07/0.31 % empty_iff
% 0.07/0.31 thf(fact_183_all__not__in__conv,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( ! [X3: product_prod_a_a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ X3 @ A3 ) )
% 0.07/0.31 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % all_not_in_conv
% 0.07/0.31 thf(fact_184_all__not__in__conv,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( ! [X3: a] :
% 0.07/0.31 ~ ( member_a @ X3 @ A3 ) )
% 0.07/0.31 = ( A3 = bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % all_not_in_conv
% 0.07/0.31 thf(fact_185_empty__Collect__eq,axiom,
% 0.07/0.31 ! [P: a > $o] :
% 0.07/0.31 ( ( bot_bot_set_a
% 0.07/0.31 = ( collect_a @ P ) )
% 0.07/0.31 = ( ! [X3: a] :
% 0.07/0.31 ~ ( P @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % empty_Collect_eq
% 0.07/0.31 thf(fact_186_Collect__empty__eq,axiom,
% 0.07/0.31 ! [P: a > $o] :
% 0.07/0.31 ( ( ( collect_a @ P )
% 0.07/0.31 = bot_bot_set_a )
% 0.07/0.31 = ( ! [X3: a] :
% 0.07/0.31 ~ ( P @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_empty_eq
% 0.07/0.31 thf(fact_187_subrelI,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: a,Y3: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S ) )
% 0.07/0.31 => ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 0.07/0.31
% 0.07/0.31 % subrelI
% 0.07/0.31 thf(fact_188_ex__in__conv,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 0.07/0.31 = ( A3 != bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % ex_in_conv
% 0.07/0.31 thf(fact_189_ex__in__conv,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
% 0.07/0.31 = ( A3 != bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % ex_in_conv
% 0.07/0.31 thf(fact_190_equals0I,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [Y3: product_prod_a_a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ Y3 @ A3 )
% 0.07/0.31 => ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0I
% 0.07/0.31 thf(fact_191_equals0I,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ! [Y3: a] :
% 0.07/0.31 ~ ( member_a @ Y3 @ A3 )
% 0.07/0.31 => ( A3 = bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0I
% 0.07/0.31 thf(fact_192_equals0D,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
% 0.07/0.31 ( ( A3 = bot_bo2131659635od_a_a )
% 0.07/0.31 => ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0D
% 0.07/0.31 thf(fact_193_equals0D,axiom,
% 0.07/0.31 ! [A3: set_a,A: a] :
% 0.07/0.31 ( ( A3 = bot_bot_set_a )
% 0.07/0.31 => ~ ( member_a @ A @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0D
% 0.07/0.31 thf(fact_194_emptyE,axiom,
% 0.07/0.31 ! [A: product_prod_a_a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 0.07/0.31
% 0.07/0.31 % emptyE
% 0.07/0.31 thf(fact_195_emptyE,axiom,
% 0.07/0.31 ! [A: a] :
% 0.07/0.31 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.07/0.31
% 0.07/0.31 % emptyE
% 0.07/0.31 thf(fact_196_Collect__mono__iff,axiom,
% 0.07/0.31 ! [P: a > $o,Q: a > $o] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.07/0.31 = ( ! [X3: a] :
% 0.07/0.31 ( ( P @ X3 )
% 0.07/0.31 => ( Q @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono_iff
% 0.07/0.31 thf(fact_197_set__eq__subset,axiom,
% 0.07/0.31 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.07/0.31 = ( ^ [A4: set_a,B5: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A4 @ B5 )
% 0.07/0.31 & ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % set_eq_subset
% 0.07/0.31 thf(fact_198_subset__trans,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a,C3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A3 @ C3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_trans
% 0.07/0.31 thf(fact_199_Collect__mono,axiom,
% 0.07/0.31 ! [P: a > $o,Q: a > $o] :
% 0.07/0.31 ( ! [X4: a] :
% 0.07/0.31 ( ( P @ X4 )
% 0.07/0.31 => ( Q @ X4 ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono
% 0.07/0.31 thf(fact_200_subset__refl,axiom,
% 0.07/0.31 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 0.07/0.31
% 0.07/0.31 % subset_refl
% 0.07/0.31 thf(fact_201_subset__iff,axiom,
% 0.07/0.31 ( ord_le1824328871od_a_a
% 0.07/0.31 = ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.07/0.31 ! [T: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ T @ A4 )
% 0.07/0.31 => ( member449909584od_a_a @ T @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_iff
% 0.07/0.31 thf(fact_202_subset__iff,axiom,
% 0.07/0.31 ( ord_less_eq_set_a
% 0.07/0.31 = ( ^ [A4: set_a,B5: set_a] :
% 0.07/0.31 ! [T: a] :
% 0.07/0.31 ( ( member_a @ T @ A4 )
% 0.07/0.31 => ( member_a @ T @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_iff
% 0.07/0.31 thf(fact_203_equalityD2,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( A3 = B3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD2
% 0.07/0.31 thf(fact_204_equalityD1,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( A3 = B3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD1
% 0.07/0.31 thf(fact_205_subset__eq,axiom,
% 0.07/0.31 ( ord_le1824328871od_a_a
% 0.07/0.31 = ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.07/0.31 ! [X3: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.07/0.31 => ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_eq
% 0.07/0.31 thf(fact_206_subset__eq,axiom,
% 0.07/0.31 ( ord_less_eq_set_a
% 0.07/0.31 = ( ^ [A4: set_a,B5: set_a] :
% 0.07/0.31 ! [X3: a] :
% 0.07/0.31 ( ( member_a @ X3 @ A4 )
% 0.07/0.31 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_eq
% 0.07/0.31 thf(fact_207_equalityE,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( A3 = B3 )
% 0.07/0.31 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityE
% 0.07/0.31 thf(fact_208_subsetD,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: product_prod_a_a] :
% 0.07/0.31 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 0.07/0.31 => ( ( member449909584od_a_a @ C2 @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ C2 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetD
% 0.07/0.31 thf(fact_209_subsetD,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a,C2: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ( member_a @ C2 @ A3 )
% 0.07/0.31 => ( member_a @ C2 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetD
% 0.07/0.31 thf(fact_210_in__mono,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,X2: product_prod_a_a] :
% 0.07/0.31 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 0.07/0.31 => ( ( member449909584od_a_a @ X2 @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ X2 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % in_mono
% 0.07/0.31 thf(fact_211_in__mono,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a,X2: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ( member_a @ X2 @ A3 )
% 0.07/0.31 => ( member_a @ X2 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % in_mono
% 0.07/0.31 thf(fact_212_Pair__mono,axiom,
% 0.07/0.31 ! [X2: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X2 @ X5 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 0.07/0.31 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Pair_mono
% 0.07/0.31 thf(fact_213_bot__prod__def,axiom,
% 0.07/0.31 ( bot_bo1519632275_set_a
% 0.07/0.31 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot_prod_def
% 0.07/0.31 thf(fact_214_order__refl,axiom,
% 0.07/0.31 ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% 0.07/0.31
% 0.07/0.31 % order_refl
% 0.07/0.31 thf(fact_215_subset__emptyI,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: product_prod_a_a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_emptyI
% 0.07/0.31 thf(fact_216_subset__emptyI,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ! [X4: a] :
% 0.07/0.31 ~ ( member_a @ X4 @ A3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_emptyI
% 0.07/0.31 thf(fact_217_bot_Oextremum__uniqueI,axiom,
% 0.07/0.31 ! [A: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.07/0.31 => ( A = bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot.extremum_uniqueI
% 0.07/0.31 thf(fact_218_bot_Oextremum__unique,axiom,
% 0.07/0.31 ! [A: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.07/0.31 = ( A = bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot.extremum_unique
% 0.07/0.31 thf(fact_219_bot_Oextremum,axiom,
% 0.07/0.31 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 0.07/0.31
% 0.07/0.31 % bot.extremum
% 0.07/0.31 thf(fact_220_bot__set__def,axiom,
% 0.07/0.31 ( bot_bot_set_a
% 0.07/0.31 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot_set_def
% 0.07/0.31 thf(fact_221_bot__empty__eq,axiom,
% 0.07/0.31 ( bot_bo1293121322_a_a_o
% 0.07/0.31 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot_empty_eq
% 0.07/0.31 thf(fact_222_bot__empty__eq,axiom,
% 0.07/0.31 ( bot_bot_a_o
% 0.07/0.31 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bot_empty_eq
% 0.07/0.31 thf(fact_223_order__subst1,axiom,
% 0.07/0.31 ! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ ( F @ B ) )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.07/0.31 => ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_subst1
% 0.07/0.31 thf(fact_224_order__subst2,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
% 0.07/0.31 => ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_subst2
% 0.07/0.31 thf(fact_225_ord__eq__le__subst,axiom,
% 0.07/0.31 ! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
% 0.07/0.31 ( ( A
% 0.07/0.31 = ( F @ B ) )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.07/0.31 => ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_eq_le_subst
% 0.07/0.31 thf(fact_226_ord__le__eq__subst,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( ( ( F @ B )
% 0.07/0.31 = C2 )
% 0.07/0.31 => ( ! [X4: set_a,Y3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_le_eq_subst
% 0.07/0.31 thf(fact_227_eq__iff,axiom,
% 0.07/0.31 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.07/0.31 = ( ^ [X3: set_a,Y2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.07/0.31 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_iff
% 0.07/0.31 thf(fact_228_antisym,axiom,
% 0.07/0.31 ! [X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ Y @ X2 )
% 0.07/0.31 => ( X2 = Y ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % antisym
% 0.07/0.31 thf(fact_229_eq__refl,axiom,
% 0.07/0.31 ! [X2: set_a,Y: set_a] :
% 0.07/0.31 ( ( X2 = Y )
% 0.07/0.31 => ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_refl
% 0.07/0.31 thf(fact_230_order_Otrans,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order.trans
% 0.07/0.31 thf(fact_231_antisym__conv,axiom,
% 0.07/0.31 ! [Y: set_a,X2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ Y @ X2 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.07/0.31 = ( X2 = Y ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % antisym_conv
% 0.07/0.31 thf(fact_232_order__class_Oorder_Oeq__iff,axiom,
% 0.07/0.31 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.07/0.31 = ( ^ [A6: set_a,B6: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A6 @ B6 )
% 0.07/0.31 & ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_class.order.eq_iff
% 0.07/0.31 thf(fact_233_ord__eq__le__trans,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,C2: set_a] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_eq_le_trans
% 0.07/0.31 thf(fact_234_ord__le__eq__trans,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( ( B = C2 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_le_eq_trans
% 0.07/0.31 thf(fact_235_order__class_Oorder_Oantisym,axiom,
% 0.07/0.31 ! [A: set_a,B: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ B @ A )
% 0.07/0.31 => ( A = B ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_class.order.antisym
% 0.07/0.31 thf(fact_236_order__trans,axiom,
% 0.07/0.31 ! [X2: set_a,Y: set_a,Z: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X2 @ Y )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ Y @ Z )
% 0.07/0.31 => ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_trans
% 0.07/0.31 thf(fact_237_dual__order_Orefl,axiom,
% 0.07/0.31 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.07/0.31
% 0.07/0.31 % dual_order.refl
% 0.07/0.31 thf(fact_238_dual__order_Otrans,axiom,
% 0.07/0.31 ! [B: set_a,A: set_a,C2: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ B @ A )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ C2 @ B )
% 0.07/0.31 => ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % dual_order.trans
% 0.07/0.31 thf(fact_239_dual__order_Oeq__iff,axiom,
% 0.07/0.31 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.07/0.31 = ( ^ [A6: set_a,B6: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ B6 @ A6 )
% 0.07/0.31 & ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % dual_order.eq_iff
% 0.07/0.31 thf(fact_240_dual__order_Oantisym,axiom,
% 0.07/0.31 ! [B: set_a,A: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ B @ A )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ A @ B )
% 0.07/0.31 => ( A = B ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % dual_order.antisym
% 0.07/0.31 thf(fact_241_ssubst__Pair__rhs,axiom,
% 0.07/0.31 ! [R: a,S: a,R3: set_Product_prod_a_a,S2: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R3 )
% 0.07/0.31 => ( ( S2 = S )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ssubst_Pair_rhs
% 0.07/0.31 thf(fact_242_transitivity,axiom,
% 0.07/0.31 trans_a @ relation ).
% 0.07/0.31
% 0.07/0.31 % transitivity
% 0.07/0.31 thf(fact_243_Set_Ois__empty__def,axiom,
% 0.07/0.31 ( is_empty_a
% 0.07/0.31 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Set.is_empty_def
% 0.07/0.31 thf(fact_244_transD,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( trans_a @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % transD
% 0.07/0.31 thf(fact_245_transE,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 0.07/0.31 ( ( trans_a @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % transE
% 0.07/0.31 thf(fact_246_transI,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a] :
% 0.07/0.31 ( ! [X4: a,Y3: a,Z3: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
% 0.07/0.31 => ( trans_a @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % transI
% 0.07/0.31 thf(fact_247_trans__def,axiom,
% 0.07/0.31 ( trans_a
% 0.07/0.31 = ( ^ [R2: set_Product_prod_a_a] :
% 0.07/0.31 ! [X3: a,Y2: a,Z4: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % trans_def
% 0.07/0.31 thf(fact_248_trans__empty,axiom,
% 0.07/0.31 trans_a @ bot_bo2131659635od_a_a ).
% 0.07/0.31
% 0.07/0.31 % trans_empty
% 0.07/0.31 thf(fact_249_preference_Otransitivity,axiom,
% 0.07/0.31 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.07/0.31 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.07/0.31 => ( trans_a @ Relation ) ) ).
% 0.07/0.31
% 0.07/0.31 % preference.transitivity
% 0.07/0.31 thf(fact_250_preorder__on__def,axiom,
% 0.07/0.31 ( order_preorder_on_a
% 0.07/0.31 = ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
% 0.07/0.31 ( ( refl_on_a @ A4 @ R2 )
% 0.07/0.31 & ( trans_a @ R2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % preorder_on_def
% 0.07/0.31 thf(fact_251_Collect__empty__eq__bot,axiom,
% 0.07/0.31 ! [P: a > $o] :
% 0.07/0.31 ( ( ( collect_a @ P )
% 0.07/0.31 = bot_bot_set_a )
% 0.07/0.31 = ( P = bot_bot_a_o ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_empty_eq_bot
% 0.07/0.31 thf(fact_252_relChain__def,axiom,
% 0.07/0.31 ( bNF_Ca850758276_set_a
% 0.07/0.31 = ( ^ [R2: set_Product_prod_a_a,As: a > set_a] :
% 0.07/0.31 ! [I: a,J: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % relChain_def
% 0.07/0.31 thf(fact_253_Id__on__empty,axiom,
% 0.07/0.31 ( ( id_on_a @ bot_bot_set_a )
% 0.07/0.31 = bot_bo2131659635od_a_a ) ).
% 0.07/0.31
% 0.07/0.31 % Id_on_empty
% 0.07/0.31 thf(fact_254_Id__onI,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_onI
% 0.07/0.31 thf(fact_255_Id__onI,axiom,
% 0.07/0.31 ! [A: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ A @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_onI
% 0.07/0.31 thf(fact_256_Id__on__iff,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,Y: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ ( id_on_1182990020od_a_a @ A3 ) )
% 0.07/0.31 = ( ( X2 = Y )
% 0.07/0.31 & ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_on_iff
% 0.07/0.31 thf(fact_257_Id__on__iff,axiom,
% 0.07/0.31 ! [X2: a,Y: a,A3: set_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( id_on_a @ A3 ) )
% 0.07/0.31 = ( ( X2 = Y )
% 0.07/0.31 & ( member_a @ X2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_on_iff
% 0.07/0.31 thf(fact_258_Id__on__eqI,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_on_eqI
% 0.07/0.31 thf(fact_259_Id__on__eqI,axiom,
% 0.07/0.31 ! [A: a,B: a,A3: set_a] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( ( member_a @ A @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_on_eqI
% 0.07/0.31 thf(fact_260_Id__onE,axiom,
% 0.07/0.31 ! [C2: produc1572603623od_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member2057358096od_a_a @ C2 @ ( id_on_1182990020od_a_a @ A3 ) )
% 0.07/0.31 => ~ ! [X4: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( C2
% 0.07/0.31 != ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_onE
% 0.07/0.31 thf(fact_261_Id__onE,axiom,
% 0.07/0.31 ! [C2: product_prod_a_a,A3: set_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ C2 @ ( id_on_a @ A3 ) )
% 0.07/0.31 => ~ ! [X4: a] :
% 0.07/0.31 ( ( member_a @ X4 @ A3 )
% 0.07/0.31 => ( C2
% 0.07/0.31 != ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Id_onE
% 0.07/0.31 thf(fact_262_trans__Id__on,axiom,
% 0.07/0.31 ! [A3: set_a] : ( trans_a @ ( id_on_a @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % trans_Id_on
% 0.07/0.31 thf(fact_263_refl__on__Id__on,axiom,
% 0.07/0.31 ! [A3: set_a] : ( refl_on_a @ A3 @ ( id_on_a @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_Id_on
% 0.07/0.31 thf(fact_264_under__incr,axiom,
% 0.07/0.31 ! [R: set_Product_prod_a_a,A: a,B: a] :
% 0.07/0.31 ( ( trans_a @ R )
% 0.07/0.31 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( order_under_a @ R @ A ) @ ( order_under_a @ R @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % under_incr
% 0.07/0.31 thf(fact_265_GreatestI2__order,axiom,
% 0.07/0.31 ! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
% 0.07/0.31 ( ( P @ X2 )
% 0.07/0.31 => ( ! [Y3: set_a] :
% 0.07/0.31 ( ( P @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ Y3 @ X2 ) )
% 0.07/0.31 => ( ! [X4: set_a] :
% 0.07/0.31 ( ( P @ X4 )
% 0.07/0.31 => ( ! [Y6: set_a] :
% 0.07/0.31 ( ( P @ Y6 )
% 0.07/0.31 => ( ord_less_eq_set_a @ Y6 @ X4 ) )
% 0.07/0.31 => ( Q @ X4 ) ) )
% 0.07/0.31 => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % GreatestI2_order
% 0.07/0.31 thf(fact_266_Greatest__equality,axiom,
% 0.07/0.31 ! [P: set_a > $o,X2: set_a] :
% 0.07/0.31 ( ( P @ X2 )
% 0.07/0.31 => ( ! [Y3: set_a] :
% 0.07/0.31 ( ( P @ Y3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ Y3 @ X2 ) )
% 0.07/0.31 => ( ( order_Greatest_set_a @ P )
% 0.07/0.31 = X2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Greatest_equality
% 0.07/0.31 thf(fact_267_trans__singleton,axiom,
% 0.07/0.31 ! [A: a] : ( trans_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ A ) @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % trans_singleton
% 0.07/0.31 thf(fact_268_insert__iff,axiom,
% 0.07/0.31 ! [A: a,B: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.07/0.31 = ( ( A = B )
% 0.07/0.31 | ( member_a @ A @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_iff
% 0.07/0.31 thf(fact_269_insert__iff,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 0.07/0.31 = ( ( A = B )
% 0.07/0.31 | ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_iff
% 0.07/0.31 thf(fact_270_insertCI,axiom,
% 0.07/0.31 ! [A: a,B3: set_a,B: a] :
% 0.07/0.31 ( ( ~ ( member_a @ A @ B3 )
% 0.07/0.31 => ( A = B ) )
% 0.07/0.31 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertCI
% 0.07/0.31 thf(fact_271_insertCI,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.07/0.31 ( ( ~ ( member449909584od_a_a @ A @ B3 )
% 0.07/0.31 => ( A = B ) )
% 0.07/0.31 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertCI
% 0.07/0.31 thf(fact_272_singletonI,axiom,
% 0.07/0.31 ! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonI
% 0.07/0.31 thf(fact_273_singletonI,axiom,
% 0.07/0.31 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonI
% 0.07/0.31 thf(fact_274_insert__subset,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ A3 ) @ B3 )
% 0.07/0.31 = ( ( member449909584od_a_a @ X2 @ B3 )
% 0.07/0.31 & ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_subset
% 0.07/0.31 thf(fact_275_insert__subset,axiom,
% 0.07/0.31 ! [X2: a,A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
% 0.07/0.31 = ( ( member_a @ X2 @ B3 )
% 0.07/0.31 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_subset
% 0.07/0.31 thf(fact_276_singleton__insert__inj__eq,axiom,
% 0.07/0.31 ! [B: a,A: a,A3: set_a] :
% 0.07/0.31 ( ( ( insert_a @ B @ bot_bot_set_a )
% 0.07/0.31 = ( insert_a @ A @ A3 ) )
% 0.07/0.31 = ( ( A = B )
% 0.07/0.31 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_insert_inj_eq
% 0.07/0.31 thf(fact_277_singleton__insert__inj__eq_H,axiom,
% 0.07/0.31 ! [A: a,A3: set_a,B: a] :
% 0.07/0.31 ( ( ( insert_a @ A @ A3 )
% 0.07/0.31 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.07/0.31 = ( ( A = B )
% 0.07/0.31 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_insert_inj_eq'
% 0.07/0.31 thf(fact_278_prod_Ocollapse,axiom,
% 0.07/0.31 ! [Prod: product_prod_a_a] :
% 0.07/0.31 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 0.07/0.31 = Prod ) ).
% 0.07/0.31
% 0.07/0.31 % prod.collapse
% 0.07/0.31 thf(fact_279_insert__subsetI,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,X6: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X2 @ A3 )
% 0.07/0.31 => ( ( ord_le1824328871od_a_a @ X6 @ A3 )
% 0.07/0.31 => ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_subsetI
% 0.07/0.31 thf(fact_280_insert__subsetI,axiom,
% 0.07/0.31 ! [X2: a,A3: set_a,X6: set_a] :
% 0.07/0.31 ( ( member_a @ X2 @ A3 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ X6 @ A3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( insert_a @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_subsetI
% 0.07/0.31 thf(fact_281_less__eq__prod__def,axiom,
% 0.07/0.31 ( ord_le486764743_set_a
% 0.07/0.31 = ( ^ [X3: produc1691597095_set_a,Y2: produc1691597095_set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ ( produc1926412547_set_a @ X3 ) @ ( produc1926412547_set_a @ Y2 ) )
% 0.07/0.31 & ( ord_less_eq_set_a @ ( produc918972485_set_a @ X3 ) @ ( produc918972485_set_a @ Y2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % less_eq_prod_def
% 0.07/0.31 thf(fact_282_singleton__inject,axiom,
% 0.07/0.31 ! [A: a,B: a] :
% 0.07/0.31 ( ( ( insert_a @ A @ bot_bot_set_a )
% 0.07/0.31 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.07/0.31 => ( A = B ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_inject
% 0.07/0.31 thf(fact_283_insert__not__empty,axiom,
% 0.07/0.31 ! [A: a,A3: set_a] :
% 0.07/0.31 ( ( insert_a @ A @ A3 )
% 0.07/0.31 != bot_bot_set_a ) ).
% 0.07/0.31
% 0.07/0.31 % insert_not_empty
% 0.07/0.31 thf(fact_284_doubleton__eq__iff,axiom,
% 0.07/0.31 ! [A: a,B: a,C2: a,D: a] :
% 0.07/0.31 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 0.07/0.31 = ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 0.07/0.31 = ( ( ( A = C2 )
% 0.07/0.31 & ( B = D ) )
% 0.07/0.31 | ( ( A = D )
% 0.07/0.31 & ( B = C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % doubleton_eq_iff
% 0.07/0.31 thf(fact_285_singleton__iff,axiom,
% 0.07/0.31 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 0.07/0.31 = ( B = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_iff
% 0.07/0.31 thf(fact_286_singleton__iff,axiom,
% 0.07/0.31 ! [B: a,A: a] :
% 0.07/0.31 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.07/0.31 = ( B = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_iff
% 0.07/0.31 thf(fact_287_singletonD,axiom,
% 0.07/0.31 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 0.07/0.31 => ( B = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonD
% 0.07/0.31 thf(fact_288_singletonD,axiom,
% 0.07/0.31 ! [B: a,A: a] :
% 0.07/0.31 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.07/0.31 => ( B = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonD
% 0.07/0.31 thf(fact_289_subset__insertI2,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a,B: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insertI2
% 0.07/0.31 thf(fact_290_subset__insertI,axiom,
% 0.07/0.31 ! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insertI
% 0.07/0.31 thf(fact_291_subset__insert,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.07/0.31 ( ~ ( member449909584od_a_a @ X2 @ A3 )
% 0.07/0.31 => ( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X2 @ B3 ) )
% 0.07/0.31 = ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insert
% 0.07/0.31 thf(fact_292_subset__insert,axiom,
% 0.07/0.31 ! [X2: a,A3: set_a,B3: set_a] :
% 0.07/0.31 ( ~ ( member_a @ X2 @ A3 )
% 0.07/0.31 => ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
% 0.07/0.31 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insert
% 0.07/0.31 thf(fact_293_insert__mono,axiom,
% 0.07/0.31 ! [C3: set_a,D2: set_a,A: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ C3 @ D2 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_mono
% 0.07/0.31 thf(fact_294_mk__disjoint__insert,axiom,
% 0.07/0.31 ! [A: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ A @ A3 )
% 0.07/0.31 => ? [B7: set_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert_a @ A @ B7 ) )
% 0.07/0.31 & ~ ( member_a @ A @ B7 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mk_disjoint_insert
% 0.07/0.31 thf(fact_295_mk__disjoint__insert,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ? [B7: set_Product_prod_a_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert1116662519od_a_a @ A @ B7 ) )
% 0.07/0.31 & ~ ( member449909584od_a_a @ A @ B7 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mk_disjoint_insert
% 0.07/0.31 thf(fact_296_insert__eq__iff,axiom,
% 0.07/0.31 ! [A: a,A3: set_a,B: a,B3: set_a] :
% 0.07/0.31 ( ~ ( member_a @ A @ A3 )
% 0.07/0.31 => ( ~ ( member_a @ B @ B3 )
% 0.07/0.31 => ( ( ( insert_a @ A @ A3 )
% 0.07/0.31 = ( insert_a @ B @ B3 ) )
% 0.07/0.31 = ( ( ( A = B )
% 0.07/0.31 => ( A3 = B3 ) )
% 0.07/0.31 & ( ( A != B )
% 0.07/0.31 => ? [C4: set_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert_a @ B @ C4 ) )
% 0.07/0.31 & ~ ( member_a @ B @ C4 )
% 0.07/0.31 & ( B3
% 0.07/0.31 = ( insert_a @ A @ C4 ) )
% 0.07/0.31 & ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_eq_iff
% 0.07/0.31 thf(fact_297_insert__eq__iff,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.07/0.31 ( ~ ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ( ~ ( member449909584od_a_a @ B @ B3 )
% 0.07/0.31 => ( ( ( insert1116662519od_a_a @ A @ A3 )
% 0.07/0.31 = ( insert1116662519od_a_a @ B @ B3 ) )
% 0.07/0.31 = ( ( ( A = B )
% 0.07/0.31 => ( A3 = B3 ) )
% 0.07/0.31 & ( ( A != B )
% 0.07/0.31 => ? [C4: set_Product_prod_a_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert1116662519od_a_a @ B @ C4 ) )
% 0.07/0.31 & ~ ( member449909584od_a_a @ B @ C4 )
% 0.07/0.31 & ( B3
% 0.07/0.31 = ( insert1116662519od_a_a @ A @ C4 ) )
% 0.07/0.31 & ~ ( member449909584od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_eq_iff
% 0.07/0.31 thf(fact_298_insert__absorb,axiom,
% 0.07/0.31 ! [A: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ A @ A3 )
% 0.07/0.31 => ( ( insert_a @ A @ A3 )
% 0.07/0.31 = A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_absorb
% 0.07/0.31 thf(fact_299_insert__absorb,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ A3 )
% 0.07/0.31 => ( ( insert1116662519od_a_a @ A @ A3 )
% 0.07/0.31 = A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_absorb
% 0.07/0.31 thf(fact_300_insert__ident,axiom,
% 0.07/0.31 ! [X2: a,A3: set_a,B3: set_a] :
% 0.07/0.31 ( ~ ( member_a @ X2 @ A3 )
% 0.07/0.31 => ( ~ ( member_a @ X2 @ B3 )
% 0.07/0.31 => ( ( ( insert_a @ X2 @ A3 )
% 0.07/0.31 = ( insert_a @ X2 @ B3 ) )
% 0.07/0.31 = ( A3 = B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_ident
% 0.07/0.31 thf(fact_301_insert__ident,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.07/0.31 ( ~ ( member449909584od_a_a @ X2 @ A3 )
% 0.07/0.31 => ( ~ ( member449909584od_a_a @ X2 @ B3 )
% 0.07/0.31 => ( ( ( insert1116662519od_a_a @ X2 @ A3 )
% 0.07/0.31 = ( insert1116662519od_a_a @ X2 @ B3 ) )
% 0.07/0.31 = ( A3 = B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_ident
% 0.07/0.31 thf(fact_302_Set_Oset__insert,axiom,
% 0.07/0.31 ! [X2: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ X2 @ A3 )
% 0.07/0.31 => ~ ! [B7: set_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert_a @ X2 @ B7 ) )
% 0.07/0.31 => ( member_a @ X2 @ B7 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Set.set_insert
% 0.07/0.31 thf(fact_303_Set_Oset__insert,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X2 @ A3 )
% 0.07/0.31 => ~ ! [B7: set_Product_prod_a_a] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( insert1116662519od_a_a @ X2 @ B7 ) )
% 0.07/0.31 => ( member449909584od_a_a @ X2 @ B7 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Set.set_insert
% 0.07/0.31 thf(fact_304_insertI2,axiom,
% 0.07/0.31 ! [A: a,B3: set_a,B: a] :
% 0.07/0.31 ( ( member_a @ A @ B3 )
% 0.07/0.31 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertI2
% 0.07/0.31 thf(fact_305_insertI2,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ B3 )
% 0.07/0.31 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertI2
% 0.07/0.31 thf(fact_306_insertI1,axiom,
% 0.07/0.31 ! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertI1
% 0.07/0.31 thf(fact_307_insertI1,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B3: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertI1
% 0.07/0.31 thf(fact_308_insertE,axiom,
% 0.07/0.31 ! [A: a,B: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.07/0.31 => ( ( A != B )
% 0.07/0.31 => ( member_a @ A @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertE
% 0.07/0.31 thf(fact_309_insertE,axiom,
% 0.07/0.31 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 0.07/0.31 => ( ( A != B )
% 0.07/0.31 => ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insertE
% 0.07/0.31 thf(fact_310_surjective__pairing,axiom,
% 0.07/0.31 ! [T2: product_prod_a_a] :
% 0.07/0.31 ( T2
% 0.07/0.31 = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surjective_pairing
% 0.07/0.31 thf(fact_311_prod_Oexhaust__sel,axiom,
% 0.07/0.31 ! [Prod: product_prod_a_a] :
% 0.07/0.31 ( Prod
% 0.07/0.31 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod.exhaust_sel
% 0.07/0.31 thf(fact_312_fst__conv,axiom,
% 0.07/0.31 ! [X1: a,X22: a] :
% 0.07/0.31 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.07/0.31 = X1 ) ).
% 0.07/0.31
% 0.07/0.31 % fst_conv
% 0.07/0.31 thf(fact_313_fst__eqD,axiom,
% 0.07/0.31 ! [X2: a,Y: a,A: a] :
% 0.07/0.31 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 0.07/0.31 = A )
% 0.07/0.31 => ( X2 = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % fst_eqD
% 0.07/0.31 thf(fact_314_snd__conv,axiom,
% 0.07/0.31 ! [X1: a,X22: a] :
% 0.07/0.31 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.07/0.31 = X22 ) ).
% 0.07/0.31
% 0.07/0.31 % snd_conv
% 0.07/0.31 thf(fact_315_snd__eqD,axiom,
% 0.07/0.31 ! [X2: a,Y: a,A: a] :
% 0.07/0.31 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 0.07/0.31 = A )
% 0.07/0.31 => ( Y = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % snd_eqD
% 0.07/0.31 thf(fact_316_refl__on__singleton,axiom,
% 0.07/0.31 ! [X2: a] : ( refl_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % refl_on_singleton
% 0.07/0.31 thf(fact_317_total__on__singleton,axiom,
% 0.07/0.31 ! [X2: a] : ( total_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % total_on_singleton
% 0.07/0.31 thf(fact_318_subset__singletonD,axiom,
% 0.07/0.31 ! [A3: set_a,X2: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) )
% 0.07/0.31 => ( ( A3 = bot_bot_set_a )
% 0.07/0.31 | ( A3
% 0.07/0.31 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_singletonD
% 0.07/0.31 thf(fact_319_subset__singleton__iff,axiom,
% 0.07/0.31 ! [X6: set_a,A: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.07/0.31 = ( ( X6 = bot_bot_set_a )
% 0.07/0.31 | ( X6
% 0.07/0.31 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_singleton_iff
% 0.07/0.31 thf(fact_320_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.07/0.31 ! [P: a > a > $o,X2: a,Y: a,A: product_prod_a_a] :
% 0.07/0.31 ( ( P @ X2 @ Y )
% 0.07/0.31 => ( ( A
% 0.07/0.31 = ( product_Pair_a_a @ X2 @ Y ) )
% 0.07/0.31 => ( P @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % BNF_Greatest_Fixpoint.subst_Pair
% 0.07/0.31 thf(fact_321_conjI__realizer,axiom,
% 0.07/0.31 ! [P: a > $o,P2: a,Q: a > $o,Q2: a] :
% 0.07/0.31 ( ( P @ P2 )
% 0.07/0.31 => ( ( Q @ Q2 )
% 0.07/0.31 => ( ( P @ ( product_fst_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) )
% 0.07/0.31 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % conjI_realizer
% 0.07/0.31 thf(fact_322_exI__realizer,axiom,
% 0.07/0.31 ! [P: a > a > $o,Y: a,X2: a] :
% 0.07/0.31 ( ( P @ Y @ X2 )
% 0.07/0.31 => ( P @ ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % exI_realizer
% 0.07/0.31 thf(fact_323_the__elem__eq,axiom,
% 0.07/0.31 ! [X2: a] :
% 0.07/0.31 ( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
% 0.07/0.31 = X2 ) ).
% 0.07/0.31
% 0.07/0.31 % the_elem_eq
% 0.07/0.31 thf(fact_324_is__singletonI,axiom,
% 0.07/0.31 ! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singletonI
% 0.07/0.31 thf(fact_325_is__singleton__the__elem,axiom,
% 0.07/0.31 ( is_singleton_a
% 0.07/0.31 = ( ^ [A4: set_a] :
% 0.07/0.31 ( A4
% 0.07/0.31 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singleton_the_elem
% 0.07/0.31 thf(fact_326_is__singletonI_H,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( A3 != bot_bo2131659635od_a_a )
% 0.07/0.31 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member449909584od_a_a @ Y3 @ A3 )
% 0.07/0.31 => ( X4 = Y3 ) ) )
% 0.07/0.31 => ( is_sin877026331od_a_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singletonI'
% 0.07/0.31 thf(fact_327_is__singletonI_H,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( A3 != bot_bot_set_a )
% 0.07/0.31 => ( ! [X4: a,Y3: a] :
% 0.07/0.31 ( ( member_a @ X4 @ A3 )
% 0.07/0.31 => ( ( member_a @ Y3 @ A3 )
% 0.07/0.31 => ( X4 = Y3 ) ) )
% 0.07/0.31 => ( is_singleton_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singletonI'
% 0.07/0.31 thf(fact_328_is__singletonE,axiom,
% 0.07/0.31 ! [A3: set_a] :
% 0.07/0.31 ( ( is_singleton_a @ A3 )
% 0.07/0.31 => ~ ! [X4: a] :
% 0.07/0.31 ( A3
% 0.07/0.31 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singletonE
% 0.07/0.31 thf(fact_329_is__singleton__def,axiom,
% 0.07/0.31 ( is_singleton_a
% 0.07/0.31 = ( ^ [A4: set_a] :
% 0.07/0.31 ? [X3: a] :
% 0.07/0.31 ( A4
% 0.07/0.31 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % is_singleton_def
% 0.07/0.31 thf(fact_330_linear__order__on__singleton,axiom,
% 0.07/0.31 ! [X2: a] : ( order_653178931r_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 0.07/0.31
% 0.07/0.31 % linear_order_on_singleton
% 0.07/0.31 thf(fact_331_sndI,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,Y: a,Z: a] :
% 0.07/0.31 ( ( X2
% 0.07/0.31 = ( product_Pair_a_a @ Y @ Z ) )
% 0.07/0.31 => ( ( product_snd_a_a @ X2 )
% 0.07/0.31 = Z ) ) ).
% 0.07/0.31
% 0.07/0.31 % sndI
% 0.07/0.31 thf(fact_332_lnear__order__on__empty,axiom,
% 0.07/0.31 order_653178931r_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.07/0.31
% 0.07/0.31 % lnear_order_on_empty
% 0.07/0.31 thf(fact_333_fstI,axiom,
% 0.07/0.31 ! [X2: product_prod_a_a,Y: a,Z: a] :
% 0.07/0.31 ( ( X2
% 0.07/0.31 = ( product_Pair_a_a @ Y @ Z ) )
% 0.07/0.31 => ( ( product_fst_a_a @ X2 )
% 0.07/0.31 = Y ) ) ).
% 0.07/0.31
% 0.07/0.31 % fstI
% 0.07/0.31 thf(fact_334_eq__snd__iff,axiom,
% 0.07/0.31 ! [B: a,P2: product_prod_a_a] :
% 0.07/0.31 ( ( B
% 0.07/0.31 = ( product_snd_a_a @ P2 ) )
% 0.07/0.31 = ( ? [A6: a] :
% 0.07/0.31 ( P2
% 0.07/0.31 = ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_snd_iff
% 0.07/0.31 thf(fact_335_eq__fst__iff,axiom,
% 0.07/0.31 ! [A: a,P2: product_prod_a_a] :
% 0.07/0.31 ( ( A
% 0.07/0.31 = ( product_fst_a_a @ P2 ) )
% 0.07/0.31 = ( ? [B6: a] :
% 0.07/0.31 ( P2
% 0.07/0.31 = ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_fst_iff
% 0.07/0.31 thf(fact_336_prod_Oswap__def,axiom,
% 0.07/0.31 ( product_swap_a_a
% 0.07/0.31 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % prod.swap_def
% 0.07/0.31 thf(fact_337_Range__insert,axiom,
% 0.07/0.31 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( range_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
% 0.07/0.31 = ( insert_a @ B @ ( range_a_a @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range_insert
% 0.07/0.31 thf(fact_338_swap__simp,axiom,
% 0.07/0.31 ! [X2: a,Y: a] :
% 0.07/0.31 ( ( product_swap_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 0.07/0.31 = ( product_Pair_a_a @ Y @ X2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % swap_simp
% 0.07/0.31 thf(fact_339_RangeE,axiom,
% 0.07/0.31 ! [B: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ B @ ( range_a_a @ R ) )
% 0.07/0.31 => ~ ! [A5: a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B ) @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % RangeE
% 0.07/0.31 thf(fact_340_Range__iff,axiom,
% 0.07/0.31 ! [A: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ A @ ( range_a_a @ R ) )
% 0.07/0.31 = ( ? [Y2: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ A ) @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range_iff
% 0.07/0.31 thf(fact_341_Range_Ocases,axiom,
% 0.07/0.31 ! [A: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ A @ ( range_a_a @ R ) )
% 0.07/0.31 => ~ ! [A5: a] :
% 0.07/0.31 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ A ) @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range.cases
% 0.07/0.31 thf(fact_342_Range_Osimps,axiom,
% 0.07/0.31 ! [A: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member_a @ A @ ( range_a_a @ R ) )
% 0.07/0.31 = ( ? [A6: a,B6: a] :
% 0.07/0.31 ( ( A = B6 )
% 0.07/0.31 & ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ R ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range.simps
% 0.07/0.31 thf(fact_343_Range_Ointros,axiom,
% 0.07/0.31 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 0.07/0.31 => ( member_a @ B @ ( range_a_a @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range.intros
% 0.07/0.31 thf(fact_344_Range_Oinducts,axiom,
% 0.07/0.31 ! [X2: a,R: set_Product_prod_a_a,P: a > $o] :
% 0.07/0.31 ( ( member_a @ X2 @ ( range_a_a @ R ) )
% 0.07/0.31 => ( ! [A5: a,B4: a] :
% 0.07/0.31 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B4 ) @ R )
% 0.07/0.31 => ( P @ B4 ) )
% 0.07/0.31 => ( P @ X2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Range.inducts
% 0.07/0.31 thf(fact_345_Domain__insert,axiom,
% 0.07/0.31 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 0.07/0.31 ( ( domain_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
% 0.07/0.31 = ( insert_a @ A @ ( domain_a_a @ R ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Domain_insert
% 0.07/0.31 thf(fact_346_subset__Compl__singleton,axiom,
% 0.07/0.31 ! [A3: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.07/0.31 ( ( ord_le1824328871od_a_a @ A3 @ ( uminus1440301054od_a_a @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) )
% 0.07/0.31 = ( ~ ( member449909584od_a_a @ B @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_Compl_singleton
% 0.07/0.31 thf(fact_347_subset__Compl__singleton,axiom,
% 0.07/0.31 ! [A3: set_a,B: a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
% 0.07/0.31 = ( ~ ( member_a @ B @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_Compl_singleton
% 0.07/0.31 thf(fact_348_Compl__iff,axiom,
% 0.07/0.31 ! [C2: a,A3: set_a] :
% 0.07/0.31 ( ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) )
% 0.07/0.31 = ( ~ ( member_a @ C2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Compl_iff
% 0.07/0.31 thf(fact_349_Compl__iff,axiom,
% 0.07/0.31 ! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) )
% 0.07/0.31 = ( ~ ( member449909584od_a_a @ C2 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Compl_iff
% 0.07/0.31 thf(fact_350_ComplI,axiom,
% 0.07/0.31 ! [C2: a,A3: set_a] :
% 0.07/0.31 ( ~ ( member_a @ C2 @ A3 )
% 0.07/0.31 => ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ComplI
% 0.07/0.31 thf(fact_351_ComplI,axiom,
% 0.07/0.31 ! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.07/0.31 ( ~ ( member449909584od_a_a @ C2 @ A3 )
% 0.07/0.31 => ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ComplI
% 0.07/0.31 thf(fact_352_Compl__anti__mono,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.07/0.31 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Compl_anti_mono
% 0.07/0.31 thf(fact_353_Compl__subset__Compl__iff,axiom,
% 0.07/0.31 ! [A3: set_a,B3: set_a] :
% 0.07/0.31 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B3 ) )
% 0.07/0.31 = ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % Compl_subset_Compl_iff
% 0.07/0.31 thf(fact_354_subset__Compl__self__eq,axiom,
% 0.07/0.31 ! [A3: set_a] :
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% 2.17/1.12 thf(sy_c_Relation_Orefl__on_001t__Set__Oset_Itf__a_J,type,
% 2.17/1.12 refl_on_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Orefl__on_001tf__a,type,
% 2.17/1.12 refl_on_a: set_a > set_Product_prod_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.17/1.12 total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.17/1.12 total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Ototal__on_001t__Set__Oset_Itf__a_J,type,
% 2.17/1.12 total_on_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Ototal__on_001tf__a,type,
% 2.17/1.12 total_on_a: set_a > set_Product_prod_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Relation_Otrans_001tf__a,type,
% 2.17/1.12 trans_a: set_Product_prod_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.17/1.12 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.17/1.12 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.17/1.12 collect_a: ( a > $o ) > set_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.17/1.12 insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Oinsert_001tf__a,type,
% 2.17/1.12 insert_a: a > set_a > set_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 2.17/1.12 is_empty_a: set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.17/1.12 is_sin877026331od_a_a: set_Product_prod_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Ois__singleton_001tf__a,type,
% 2.17/1.12 is_singleton_a: set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_Set_Othe__elem_001tf__a,type,
% 2.17/1.12 the_elem_a: set_a > a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.17/1.12 member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.17/1.12 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.17/1.12 member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.17/1.12 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 2.17/1.12 member_set_a: set_a > set_set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_c_member_001tf__a,type,
% 2.17/1.12 member_a: a > set_a > $o ).
% 2.17/1.12
% 2.17/1.12 thf(sy_v_carrier,type,
% 2.17/1.12 carrier: set_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_v_relation,type,
% 2.17/1.12 relation: set_Product_prod_a_a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_v_y,type,
% 2.17/1.12 y: a ).
% 2.17/1.12
% 2.17/1.12 thf(sy_v_z,type,
% 2.17/1.12 z: a ).
% 2.17/1.12
% 2.17/1.12 thf(fact_0_assms,axiom,
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ y @ z ) @ relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_1_rational__preference__axioms,axiom,
% 2.17/1.12 prefer719835986ence_a @ carrier @ relation ).
% 2.17/1.12
% 2.17/1.12 thf(fact_2_preference__axioms,axiom,
% 2.17/1.12 prefer1292084992ence_a @ carrier @ relation ).
% 2.17/1.12
% 2.17/1.12 thf(fact_3_trans__refl,axiom,
% 2.17/1.12 order_preorder_on_a @ carrier @ relation ).
% 2.17/1.12
% 2.17/1.12 thf(fact_4_total,axiom,
% 2.17/1.12 total_on_a @ carrier @ relation ).
% 2.17/1.12
% 2.17/1.12 thf(fact_5_reflexivity,axiom,
% 2.17/1.12 refl_on_a @ carrier @ relation ).
% 2.17/1.12
% 2.17/1.12 thf(fact_6_compl,axiom,
% 2.17/1.12 ! [X: a] :
% 2.17/1.12 ( ( member_a @ X @ carrier )
% 2.17/1.12 => ! [Xa: a] :
% 2.17/1.12 ( ( member_a @ Xa @ carrier )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ relation )
% 2.17/1.12 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_7_completeD,axiom,
% 2.17/1.12 ! [X2: a,Y: a] :
% 2.17/1.12 ( ( member_a @ X2 @ carrier )
% 2.17/1.12 => ( ( member_a @ Y @ carrier )
% 2.17/1.12 => ( ( X2 != Y )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_8_not__outside,axiom,
% 2.17/1.12 ! [X2: a,Y: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 => ( member_a @ X2 @ carrier ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_9_strict__is__neg__transitive,axiom,
% 2.17/1.12 ! [X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( ( member_a @ X2 @ carrier )
% 2.17/1.12 & ( member_a @ Y @ carrier )
% 2.17/1.12 & ( member_a @ Z @ carrier ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) )
% 2.17/1.12 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_10_weak__is__transitive,axiom,
% 2.17/1.12 ! [X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( ( member_a @ X2 @ carrier )
% 2.17/1.12 & ( member_a @ Y @ carrier )
% 2.17/1.12 & ( member_a @ Z @ carrier ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.17/1.12 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_11_strict__not__refl__weak,axiom,
% 2.17/1.12 ! [X2: a,Y: a] :
% 2.17/1.12 ( ( ( member_a @ X2 @ carrier )
% 2.17/1.12 & ( member_a @ Y @ carrier ) )
% 2.17/1.12 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 2.17/1.12 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_12_indiff__trans,axiom,
% 2.17/1.12 ! [X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_13_strict__trans,axiom,
% 2.17/1.12 ! [X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_14_pref__in__at__least__as,axiom,
% 2.17/1.12 ! [X2: a,Y: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 => ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_15_rational__preference_Oaxioms_I1_J,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_16_rational__preference_Oaxioms_I1_J,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_17_preference__def,axiom,
% 2.17/1.12 ( prefer1409946857od_a_a
% 2.17/1.12 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 2.17/1.12 ( ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 2.17/1.12 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member2057358096od_a_a @ X3 @ Carrier2 ) )
% 2.17/1.12 & ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
% 2.17/1.12 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member2057358096od_a_a @ Y2 @ Carrier2 ) )
% 2.17/1.12 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_18_preference__def,axiom,
% 2.17/1.12 ( prefer1710114912_set_a
% 2.17/1.12 = ( ^ [Carrier2: set_set_a,Relation2: set_Pr2045688071_set_a] :
% 2.17/1.12 ( ! [X3: set_a,Y2: set_a] :
% 2.17/1.12 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member_set_a @ X3 @ Carrier2 ) )
% 2.17/1.12 & ! [X3: set_a,Y2: set_a] :
% 2.17/1.12 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member_set_a @ Y2 @ Carrier2 ) )
% 2.17/1.12 & ( order_816746613_set_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_19_preference__def,axiom,
% 2.17/1.12 ( prefer1338892073od_a_a
% 2.17/1.12 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 2.17/1.12 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 2.17/1.12 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 2.17/1.12 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 2.17/1.12 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_20_preference__def,axiom,
% 2.17/1.12 ( prefer1292084992ence_a
% 2.17/1.12 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.17/1.12 ( ! [X3: a,Y2: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member_a @ X3 @ Carrier2 ) )
% 2.17/1.12 & ! [X3: a,Y2: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 2.17/1.12 => ( member_a @ Y2 @ Carrier2 ) )
% 2.17/1.12 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_21_preference_Ointro,axiom,
% 2.17/1.12 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.17/1.12 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member2057358096od_a_a @ X4 @ Carrier ) )
% 2.17/1.12 => ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.17/1.12 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 2.17/1.12 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_22_preference_Ointro,axiom,
% 2.17/1.12 ! [Relation: set_Pr2045688071_set_a,Carrier: set_set_a] :
% 2.17/1.12 ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.12 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member_set_a @ X4 @ Carrier ) )
% 2.17/1.12 => ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.12 ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member_set_a @ Y3 @ Carrier ) )
% 2.17/1.12 => ( ( order_816746613_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1710114912_set_a @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_23_preference_Ointro,axiom,
% 2.17/1.12 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 2.17/1.12 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.17/1.12 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 2.17/1.12 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.17/1.12 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 2.17/1.12 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_24_preference_Ointro,axiom,
% 2.17/1.12 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 2.17/1.12 ( ! [X4: a,Y3: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member_a @ X4 @ Carrier ) )
% 2.17/1.12 => ( ! [X4: a,Y3: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 2.17/1.12 => ( member_a @ Y3 @ Carrier ) )
% 2.17/1.12 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 2.17/1.12 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_25_preference_Otrans__refl,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_26_preference_Otrans__refl,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.12 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_27_preference_Onot__outside,axiom,
% 2.17/1.12 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.12 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( member2057358096od_a_a @ X2 @ Carrier ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_28_preference_Onot__outside,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.12 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( member_set_a @ X2 @ Carrier ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_29_preference_Onot__outside,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.12 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( member449909584od_a_a @ X2 @ Carrier ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_30_preference_Onot__outside,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.12 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( member_a @ X2 @ Carrier ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_31_preference_Oreflexivity,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_32_preference_Oreflexivity,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.12 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_33_preference_Oindiff__trans,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 2.17/1.12 ( ( prefer1710114912_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_34_preference_Oindiff__trans,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.17/1.12 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_35_preference_Oindiff__trans,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_36_rational__preference_Ocompl,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ! [X: set_a] :
% 2.17/1.12 ( ( member_set_a @ X @ Carrier )
% 2.17/1.12 => ! [Xa: set_a] :
% 2.17/1.12 ( ( member_set_a @ Xa @ Carrier )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Xa ) @ Relation )
% 2.17/1.12 | ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_37_rational__preference_Ocompl,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ! [X: product_prod_a_a] :
% 2.17/1.12 ( ( member449909584od_a_a @ X @ Carrier )
% 2.17/1.12 => ! [Xa: product_prod_a_a] :
% 2.17/1.12 ( ( member449909584od_a_a @ Xa @ Carrier )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Xa ) @ Relation )
% 2.17/1.12 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_38_rational__preference_Ocompl,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ! [X: a] :
% 2.17/1.12 ( ( member_a @ X @ Carrier )
% 2.17/1.12 => ! [Xa: a] :
% 2.17/1.12 ( ( member_a @ Xa @ Carrier )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ Relation )
% 2.17/1.12 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_39_rational__preference_Ototal,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_40_rational__preference_Ototal,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( total_on_a @ Carrier @ Relation ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_41_rational__preference_OcompleteD,axiom,
% 2.17/1.12 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.12 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.12 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.17/1.12 => ( ( X2 != Y )
% 2.17/1.12 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_42_rational__preference_OcompleteD,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.12 => ( ( member_set_a @ Y @ Carrier )
% 2.17/1.12 => ( ( X2 != Y )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_43_rational__preference_OcompleteD,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.12 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.17/1.12 => ( ( X2 != Y )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_44_rational__preference_OcompleteD,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( member_a @ X2 @ Carrier )
% 2.17/1.12 => ( ( member_a @ Y @ Carrier )
% 2.17/1.12 => ( ( X2 != Y )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_45_rational__preference_Ostrict__trans,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_46_rational__preference_Ostrict__trans,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_47_rational__preference_Ostrict__trans,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_48_rational__preference_Oweak__is__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 2.17/1.12 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member2057358096od_a_a @ Y @ Carrier )
% 2.17/1.12 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_49_rational__preference_Oweak__is__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_set_a @ Y @ Carrier )
% 2.17/1.12 & ( member_set_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
% 2.17/1.12 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_50_rational__preference_Oweak__is__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member449909584od_a_a @ Y @ Carrier )
% 2.17/1.12 & ( member449909584od_a_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_51_rational__preference_Oweak__is__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_a @ Y @ Carrier )
% 2.17/1.12 & ( member_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.17/1.12 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_52_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.17/1.12 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.12 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 2.17/1.12 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_53_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_set_a @ Y @ Carrier ) )
% 2.17/1.12 => ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_54_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member449909584od_a_a @ Y @ Carrier ) )
% 2.17/1.12 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_55_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_a @ Y @ Carrier ) )
% 2.17/1.12 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_56_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
% 2.17/1.12 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member2057358096od_a_a @ Y @ Carrier )
% 2.17/1.12 & ( member2057358096od_a_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X2 ) @ Relation ) )
% 2.17/1.12 | ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_57_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
% 2.17/1.12 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_set_a @ Y @ Carrier )
% 2.17/1.12 & ( member_set_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) )
% 2.17/1.12 | ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_58_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.17/1.12 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.12 & ( member449909584od_a_a @ Y @ Carrier )
% 2.17/1.12 & ( member449909584od_a_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) )
% 2.17/1.12 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_59_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.17/1.12 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.12 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.12 => ( ( ( member_a @ X2 @ Carrier )
% 2.17/1.12 & ( member_a @ Y @ Carrier )
% 2.17/1.12 & ( member_a @ Z @ Carrier ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
% 2.17/1.12 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) )
% 2.17/1.12 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 2.17/1.12 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_60_worse__in__no__better,axiom,
% 2.17/1.12 ! [X2: a,Y: a] :
% 2.17/1.12 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.12 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_61_as__good__as__sameIff,axiom,
% 2.17/1.12 ! [Z: a,Y: a] :
% 2.17/1.12 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 2.17/1.12 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 2.17/1.12 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_62_prod_Oinject,axiom,
% 2.17/1.12 ! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y22: product_prod_a_a] :
% 2.17/1.12 ( ( ( produc1474507607od_a_a @ X1 @ X22 )
% 2.17/1.12 = ( produc1474507607od_a_a @ Y1 @ Y22 ) )
% 2.17/1.12 = ( ( X1 = Y1 )
% 2.17/1.12 & ( X22 = Y22 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_63_prod_Oinject,axiom,
% 2.17/1.12 ! [X1: set_a,X22: set_a,Y1: set_a,Y22: set_a] :
% 2.17/1.12 ( ( ( produc1928581911_set_a @ X1 @ X22 )
% 2.17/1.12 = ( produc1928581911_set_a @ Y1 @ Y22 ) )
% 2.17/1.12 = ( ( X1 = Y1 )
% 2.17/1.12 & ( X22 = Y22 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_64_prod_Oinject,axiom,
% 2.17/1.12 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 2.17/1.12 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 2.17/1.12 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 2.17/1.12 = ( ( X1 = Y1 )
% 2.17/1.12 & ( X22 = Y22 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_65_old_Oprod_Oinject,axiom,
% 2.17/1.12 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 2.17/1.12 ( ( ( produc1474507607od_a_a @ A @ B )
% 2.17/1.12 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 2.17/1.12 = ( ( A = A2 )
% 2.17/1.12 & ( B = B2 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_66_old_Oprod_Oinject,axiom,
% 2.17/1.12 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 2.17/1.12 ( ( ( produc1928581911_set_a @ A @ B )
% 2.17/1.12 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 2.17/1.12 = ( ( A = A2 )
% 2.17/1.12 & ( B = B2 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_67_old_Oprod_Oinject,axiom,
% 2.17/1.12 ! [A: a,B: a,A2: a,B2: a] :
% 2.17/1.12 ( ( ( product_Pair_a_a @ A @ B )
% 2.17/1.12 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.17/1.12 = ( ( A = A2 )
% 2.17/1.12 & ( B = B2 ) ) ) ).
% 2.17/1.12
% 2.17/1.12 thf(fact_68_rational__preference_Oas__good__as__sameIff,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,Z: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member2057358096od_a_a @ Z @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
% 2.17/1.13 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_69_rational__preference_Oas__good__as__sameIff,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member_set_a @ Z @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 = ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
% 2.17/1.13 & ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_70_rational__preference_Oas__good__as__sameIff,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 2.17/1.13 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_71_rational__preference_Oas__good__as__sameIff,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 2.17/1.13 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_72_total__onI,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a] :
% 2.17/1.13 ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.17/1.13 ( ( member2057358096od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member2057358096od_a_a @ Y3 @ A3 )
% 2.17/1.13 => ( ( X4 != Y3 )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 2.17/1.13 => ( total_1340260971od_a_a @ A3 @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_73_total__onI,axiom,
% 2.17/1.13 ! [A3: set_set_a,R: set_Pr2045688071_set_a] :
% 2.17/1.13 ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( ( member_set_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member_set_a @ Y3 @ A3 )
% 2.17/1.13 => ( ( X4 != Y3 )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 2.17/1.13 => ( total_on_set_a @ A3 @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_74_total__onI,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member449909584od_a_a @ Y3 @ A3 )
% 2.17/1.13 => ( ( X4 != Y3 )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 2.17/1.13 => ( total_1490170027od_a_a @ A3 @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_75_total__onI,axiom,
% 2.17/1.13 ! [A3: set_a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [X4: a,Y3: a] :
% 2.17/1.13 ( ( member_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member_a @ Y3 @ A3 )
% 2.17/1.13 => ( ( X4 != Y3 )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 2.17/1.13 => ( total_on_a @ A3 @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_76_total__on__def,axiom,
% 2.17/1.13 ( total_on_set_a
% 2.17/1.13 = ( ^ [A4: set_set_a,R2: set_Pr2045688071_set_a] :
% 2.17/1.13 ! [X3: set_a] :
% 2.17/1.13 ( ( member_set_a @ X3 @ A4 )
% 2.17/1.13 => ! [Y2: set_a] :
% 2.17/1.13 ( ( member_set_a @ Y2 @ A4 )
% 2.17/1.13 => ( ( X3 != Y2 )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ R2 )
% 2.17/1.13 | ( member593450832_set_a @ ( produc1928581911_set_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_77_total__on__def,axiom,
% 2.17/1.13 ( total_1490170027od_a_a
% 2.17/1.13 = ( ^ [A4: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 2.17/1.13 ! [X3: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X3 @ A4 )
% 2.17/1.13 => ! [Y2: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ Y2 @ A4 )
% 2.17/1.13 => ( ( X3 != Y2 )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ R2 )
% 2.17/1.13 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_78_total__on__def,axiom,
% 2.17/1.13 ( total_on_a
% 2.17/1.13 = ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
% 2.17/1.13 ! [X3: a] :
% 2.17/1.13 ( ( member_a @ X3 @ A4 )
% 2.17/1.13 => ! [Y2: a] :
% 2.17/1.13 ( ( member_a @ Y2 @ A4 )
% 2.17/1.13 => ( ( X3 != Y2 )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 2.17/1.13 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_79_refl__onD,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
% 2.17/1.13 ( ( refl_o198693862od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member2057358096od_a_a @ A @ A3 )
% 2.17/1.13 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_80_refl__onD,axiom,
% 2.17/1.13 ! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a] :
% 2.17/1.13 ( ( refl_on_set_a @ A3 @ R )
% 2.17/1.13 => ( ( member_set_a @ A @ A3 )
% 2.17/1.13 => ( member593450832_set_a @ ( produc1928581911_set_a @ A @ A ) @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_81_refl__onD,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 2.17/1.13 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_82_refl__onD,axiom,
% 2.17/1.13 ! [A3: set_a,R: set_Product_prod_a_a,A: a] :
% 2.17/1.13 ( ( refl_on_a @ A3 @ R )
% 2.17/1.13 => ( ( member_a @ A @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_83_refl__onD1,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( refl_o198693862od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member2057358096od_a_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_84_refl__onD1,axiom,
% 2.17/1.13 ! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( refl_on_set_a @ A3 @ R )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member_set_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_85_refl__onD1,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_86_refl__onD1,axiom,
% 2.17/1.13 ! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( refl_on_a @ A3 @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_87_refl__onD2,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( refl_o198693862od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member2057358096od_a_a @ Y @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_88_refl__onD2,axiom,
% 2.17/1.13 ! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( refl_on_set_a @ A3 @ R )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member_set_a @ Y @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_89_refl__onD2,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ Y @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_90_refl__onD2,axiom,
% 2.17/1.13 ! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( refl_on_a @ A3 @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( member_a @ Y @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_91_refl__on__domain,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
% 2.17/1.13 ( ( refl_o198693862od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R )
% 2.17/1.13 => ( ( member2057358096od_a_a @ A @ A3 )
% 2.17/1.13 & ( member2057358096od_a_a @ B @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_92_refl__on__domain,axiom,
% 2.17/1.13 ! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a,B: set_a] :
% 2.17/1.13 ( ( refl_on_set_a @ A3 @ R )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ A @ B ) @ R )
% 2.17/1.13 => ( ( member_set_a @ A @ A3 )
% 2.17/1.13 & ( member_set_a @ B @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_93_refl__on__domain,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 2.17/1.13 ( ( refl_o1298442278od_a_a @ A3 @ R )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 & ( member449909584od_a_a @ B @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_94_refl__on__domain,axiom,
% 2.17/1.13 ! [A3: set_a,R: set_Product_prod_a_a,A: a,B: a] :
% 2.17/1.13 ( ( refl_on_a @ A3 @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 2.17/1.13 => ( ( member_a @ A @ A3 )
% 2.17/1.13 & ( member_a @ B @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_95_at__least__as__goodD,axiom,
% 2.17/1.13 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.17/1.13 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_96_at__least__as__goodD,axiom,
% 2.17/1.13 ! [Z: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
% 2.17/1.13 ( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_97_at__least__as__goodD,axiom,
% 2.17/1.13 ! [Z: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_98_at__least__as__goodD,axiom,
% 2.17/1.13 ! [Z: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_99_at__lst__asgd__ge,axiom,
% 2.17/1.13 ! [X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.17/1.13 ( ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_100_at__lst__asgd__ge,axiom,
% 2.17/1.13 ! [X2: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
% 2.17/1.13 ( ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_101_at__lst__asgd__ge,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_102_at__lst__asgd__ge,axiom,
% 2.17/1.13 ! [X2: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ X2 @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Pr ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_103_old_Oprod_Oinducts,axiom,
% 2.17/1.13 ! [P: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 2.17/1.13 ( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ Prod ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_104_old_Oprod_Oinducts,axiom,
% 2.17/1.13 ! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
% 2.17/1.13 ( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ Prod ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_105_old_Oprod_Oinducts,axiom,
% 2.17/1.13 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 2.17/1.13 ( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ Prod ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_106_old_Oprod_Oexhaust,axiom,
% 2.17/1.13 ! [Y: produc1572603623od_a_a] :
% 2.17/1.13 ~ ! [A5: product_prod_a_a,B4: product_prod_a_a] :
% 2.17/1.13 ( Y
% 2.17/1.13 != ( produc1474507607od_a_a @ A5 @ B4 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_107_old_Oprod_Oexhaust,axiom,
% 2.17/1.13 ! [Y: produc1691597095_set_a] :
% 2.17/1.13 ~ ! [A5: set_a,B4: set_a] :
% 2.17/1.13 ( Y
% 2.17/1.13 != ( produc1928581911_set_a @ A5 @ B4 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_108_old_Oprod_Oexhaust,axiom,
% 2.17/1.13 ! [Y: product_prod_a_a] :
% 2.17/1.13 ~ ! [A5: a,B4: a] :
% 2.17/1.13 ( Y
% 2.17/1.13 != ( product_Pair_a_a @ A5 @ B4 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_109_mem__Collect__eq,axiom,
% 2.17/1.13 ! [A: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
% 2.17/1.13 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P ) )
% 2.17/1.13 = ( P @ A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_110_mem__Collect__eq,axiom,
% 2.17/1.13 ! [A: a,P: a > $o] :
% 2.17/1.13 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.17/1.13 = ( P @ A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_111_mem__Collect__eq,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 2.17/1.13 = ( P @ A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_112_Collect__mem__eq,axiom,
% 2.17/1.13 ! [A3: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( collec1635618130od_a_a
% 2.17/1.13 @ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A3 ) )
% 2.17/1.13 = A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_113_Collect__mem__eq,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( collect_a
% 2.17/1.13 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 2.17/1.13 = A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_114_Collect__mem__eq,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( collec645855634od_a_a
% 2.17/1.13 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 2.17/1.13 = A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_115_Collect__cong,axiom,
% 2.17/1.13 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.17/1.13 ( ! [X4: product_prod_a_a] :
% 2.17/1.13 ( ( P @ X4 )
% 2.17/1.13 = ( Q @ X4 ) )
% 2.17/1.13 => ( ( collec645855634od_a_a @ P )
% 2.17/1.13 = ( collec645855634od_a_a @ Q ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_116_Collect__cong,axiom,
% 2.17/1.13 ! [P: a > $o,Q: a > $o] :
% 2.17/1.13 ( ! [X4: a] :
% 2.17/1.13 ( ( P @ X4 )
% 2.17/1.13 = ( Q @ X4 ) )
% 2.17/1.13 => ( ( collect_a @ P )
% 2.17/1.13 = ( collect_a @ Q ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_117_prod__induct3,axiom,
% 2.17/1.13 ! [P: produc1572603623od_a_a > $o,X2: produc1572603623od_a_a] :
% 2.17/1.13 ( ! [A5: product_prod_a_a,B4: a,C: a] : ( P @ ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) )
% 2.17/1.13 => ( P @ X2 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_118_prod__cases3,axiom,
% 2.17/1.13 ! [Y: produc1572603623od_a_a] :
% 2.17/1.13 ~ ! [A5: product_prod_a_a,B4: a,C: a] :
% 2.17/1.13 ( Y
% 2.17/1.13 != ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_119_Pair__inject,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 2.17/1.13 ( ( ( produc1474507607od_a_a @ A @ B )
% 2.17/1.13 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 2.17/1.13 => ~ ( ( A = A2 )
% 2.17/1.13 => ( B != B2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_120_Pair__inject,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 2.17/1.13 ( ( ( produc1928581911_set_a @ A @ B )
% 2.17/1.13 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 2.17/1.13 => ~ ( ( A = A2 )
% 2.17/1.13 => ( B != B2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_121_Pair__inject,axiom,
% 2.17/1.13 ! [A: a,B: a,A2: a,B2: a] :
% 2.17/1.13 ( ( ( product_Pair_a_a @ A @ B )
% 2.17/1.13 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.17/1.13 => ~ ( ( A = A2 )
% 2.17/1.13 => ( B != B2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_122_prod__cases,axiom,
% 2.17/1.13 ! [P: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a] :
% 2.17/1.13 ( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ P2 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_123_prod__cases,axiom,
% 2.17/1.13 ! [P: produc1691597095_set_a > $o,P2: produc1691597095_set_a] :
% 2.17/1.13 ( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ P2 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_124_prod__cases,axiom,
% 2.17/1.13 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 2.17/1.13 ( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
% 2.17/1.13 => ( P @ P2 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_125_surj__pair,axiom,
% 2.17/1.13 ! [P2: produc1572603623od_a_a] :
% 2.17/1.13 ? [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.17/1.13 ( P2
% 2.17/1.13 = ( produc1474507607od_a_a @ X4 @ Y3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_126_surj__pair,axiom,
% 2.17/1.13 ! [P2: produc1691597095_set_a] :
% 2.17/1.13 ? [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( P2
% 2.17/1.13 = ( produc1928581911_set_a @ X4 @ Y3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_127_surj__pair,axiom,
% 2.17/1.13 ! [P2: product_prod_a_a] :
% 2.17/1.13 ? [X4: a,Y3: a] :
% 2.17/1.13 ( P2
% 2.17/1.13 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_128_rational__preference_Oworse__in__no__better,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_129_rational__preference_Oworse__in__no__better,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_130_rational__preference_Oworse__in__no__better,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_131_rational__preference_Oworse__in__no__better,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_132_rational__preference_Opref__in__at__least__as,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_133_rational__preference_Opref__in__at__least__as,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_134_rational__preference_Opref__in__at__least__as,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_135_rational__preference_Opref__in__at__least__as,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_136_at__lst__asgd__not__ge,axiom,
% 2.17/1.13 ! [X2: a,Y: a] :
% 2.17/1.13 ( ( carrier != bot_bot_set_a )
% 2.17/1.13 => ( ( member_a @ X2 @ carrier )
% 2.17/1.13 => ( ( member_a @ Y @ carrier )
% 2.17/1.13 => ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 2.17/1.13 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_137_no__better__subset__pref,axiom,
% 2.17/1.13 ! [X2: a,Y: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_138_no__better__thansubset__rel,axiom,
% 2.17/1.13 ! [X2: a,Y: a] :
% 2.17/1.13 ( ( member_a @ X2 @ carrier )
% 2.17/1.13 => ( ( member_a @ Y @ carrier )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_139_no__better__than__nonepty,axiom,
% 2.17/1.13 ! [X2: a] :
% 2.17/1.13 ( ( carrier != bot_bot_set_a )
% 2.17/1.13 => ( ( member_a @ X2 @ carrier )
% 2.17/1.13 => ( ( prefer1676310729than_a @ X2 @ carrier @ relation )
% 2.17/1.13 != bot_bot_set_a ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_140_nbt__nest,axiom,
% 2.17/1.13 ! [Y: a,X2: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
% 2.17/1.13 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_141_rational__preference_Ointro,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( prefer1064779160od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_142_rational__preference_Ointro,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.13 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_143_rational__preference__def,axiom,
% 2.17/1.13 ( prefer697821563od_a_a
% 2.17/1.13 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
% 2.17/1.13 & ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_144_rational__preference__def,axiom,
% 2.17/1.13 ( prefer719835986ence_a
% 2.17/1.13 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.17/1.13 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 2.17/1.13 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_145_refl__on__empty,axiom,
% 2.17/1.13 refl_o1298442278od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_146_refl__on__empty,axiom,
% 2.17/1.13 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_147_preorder__on__empty,axiom,
% 2.17/1.13 order_359341630od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_148_preorder__on__empty,axiom,
% 2.17/1.13 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_149_total__on__empty,axiom,
% 2.17/1.13 ! [R: set_Pr1948701895od_a_a] : ( total_1490170027od_a_a @ bot_bo2131659635od_a_a @ R ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_150_total__on__empty,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_151_rational__preference_Onbt__nest,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X2: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_152_rational__preference_Onbt__nest,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X2: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_153_rational__preference_Ono__better__than__nonepty,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bo1116205875od_a_a )
% 2.17/1.13 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( prefer926864306od_a_a @ X2 @ Carrier @ Relation )
% 2.17/1.13 != bot_bo1116205875od_a_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_154_rational__preference_Ono__better__than__nonepty,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bo2131659635od_a_a )
% 2.17/1.13 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation )
% 2.17/1.13 != bot_bo2131659635od_a_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_155_rational__preference_Ono__better__than__nonepty,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bot_set_a )
% 2.17/1.13 => ( ( member_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( prefer1676310729than_a @ X2 @ Carrier @ Relation )
% 2.17/1.13 != bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_156_rational__preference__axioms__def,axiom,
% 2.17/1.13 prefer1064779160od_a_a = total_1490170027od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_157_rational__preference__axioms__def,axiom,
% 2.17/1.13 prefer580340847ioms_a = total_on_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_158_rational__preference__axioms_Ointro,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( total_1490170027od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_159_rational__preference__axioms_Ointro,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.13 ( ( total_on_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_160_rational__preference_Oaxioms_I2_J,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_161_rational__preference_Oaxioms_I2_J,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_162_rational__preference_Ono__better__subset__pref,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_163_rational__preference_Ono__better__subset__pref,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_164_rational__preference_Ono__better__subset__pref,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_165_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.17/1.13 => ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_166_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member_set_a @ Y @ Carrier )
% 2.17/1.13 => ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 => ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_167_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.17/1.13 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_168_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( member_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member_a @ Y @ Carrier )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_169_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.17/1.13 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.17/1.13 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bo1116205875od_a_a )
% 2.17/1.13 => ( ( member2057358096od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member2057358096od_a_a @ Y @ Carrier )
% 2.17/1.13 => ( ~ ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 => ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.17/1.13 ! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( prefer484977842_set_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bot_set_set_a )
% 2.17/1.13 => ( ( member_set_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member_set_a @ Y @ Carrier )
% 2.17/1.13 => ( ~ ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 => ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_171_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.17/1.13 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
% 2.17/1.13 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bo2131659635od_a_a )
% 2.17/1.13 => ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member449909584od_a_a @ Y @ Carrier )
% 2.17/1.13 => ( ~ ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_172_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
% 2.17/1.13 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( ( Carrier != bot_bot_set_a )
% 2.17/1.13 => ( ( member_a @ X2 @ Carrier )
% 2.17/1.13 => ( ( member_a @ Y @ Carrier )
% 2.17/1.13 => ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 2.17/1.13 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_173_subset__empty,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
% 2.17/1.13 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_174_subset__empty,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 2.17/1.13 = ( A3 = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_175_empty__subsetI,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_176_empty__subsetI,axiom,
% 2.17/1.13 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_177_Pair__le,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,C2: set_a,D: set_a] :
% 2.17/1.13 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C2 @ D ) )
% 2.17/1.13 = ( ( ord_less_eq_set_a @ A @ C2 )
% 2.17/1.13 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_178_subsetI,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [X4: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ X4 @ B3 ) )
% 2.17/1.13 => ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_179_subsetI,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ! [X4: a] :
% 2.17/1.13 ( ( member_a @ X4 @ A3 )
% 2.17/1.13 => ( member_a @ X4 @ B3 ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_180_subset__antisym,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B3 @ A3 )
% 2.17/1.13 => ( A3 = B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_181_empty__iff,axiom,
% 2.17/1.13 ! [C2: product_prod_a_a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_182_empty__iff,axiom,
% 2.17/1.13 ! [C2: a] :
% 2.17/1.13 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_183_all__not__in__conv,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( ! [X3: product_prod_a_a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ X3 @ A3 ) )
% 2.17/1.13 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_184_all__not__in__conv,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( ! [X3: a] :
% 2.17/1.13 ~ ( member_a @ X3 @ A3 ) )
% 2.17/1.13 = ( A3 = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_185_empty__Collect__eq,axiom,
% 2.17/1.13 ! [P: a > $o] :
% 2.17/1.13 ( ( bot_bot_set_a
% 2.17/1.13 = ( collect_a @ P ) )
% 2.17/1.13 = ( ! [X3: a] :
% 2.17/1.13 ~ ( P @ X3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_186_Collect__empty__eq,axiom,
% 2.17/1.13 ! [P: a > $o] :
% 2.17/1.13 ( ( ( collect_a @ P )
% 2.17/1.13 = bot_bot_set_a )
% 2.17/1.13 = ( ! [X3: a] :
% 2.17/1.13 ~ ( P @ X3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_187_subrelI,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [X4: a,Y3: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S ) )
% 2.17/1.13 => ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_188_ex__in__conv,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 2.17/1.13 = ( A3 != bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_189_ex__in__conv,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
% 2.17/1.13 = ( A3 != bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_190_equals0I,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [Y3: product_prod_a_a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ Y3 @ A3 )
% 2.17/1.13 => ( A3 = bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_191_equals0I,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ! [Y3: a] :
% 2.17/1.13 ~ ( member_a @ Y3 @ A3 )
% 2.17/1.13 => ( A3 = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_192_equals0D,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
% 2.17/1.13 ( ( A3 = bot_bo2131659635od_a_a )
% 2.17/1.13 => ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_193_equals0D,axiom,
% 2.17/1.13 ! [A3: set_a,A: a] :
% 2.17/1.13 ( ( A3 = bot_bot_set_a )
% 2.17/1.13 => ~ ( member_a @ A @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_194_emptyE,axiom,
% 2.17/1.13 ! [A: product_prod_a_a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_195_emptyE,axiom,
% 2.17/1.13 ! [A: a] :
% 2.17/1.13 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_196_Collect__mono__iff,axiom,
% 2.17/1.13 ! [P: a > $o,Q: a > $o] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 2.17/1.13 = ( ! [X3: a] :
% 2.17/1.13 ( ( P @ X3 )
% 2.17/1.13 => ( Q @ X3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_197_set__eq__subset,axiom,
% 2.17/1.13 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.17/1.13 = ( ^ [A4: set_a,B5: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A4 @ B5 )
% 2.17/1.13 & ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_198_subset__trans,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a,C3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A3 @ C3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_199_Collect__mono,axiom,
% 2.17/1.13 ! [P: a > $o,Q: a > $o] :
% 2.17/1.13 ( ! [X4: a] :
% 2.17/1.13 ( ( P @ X4 )
% 2.17/1.13 => ( Q @ X4 ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_200_subset__refl,axiom,
% 2.17/1.13 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_201_subset__iff,axiom,
% 2.17/1.13 ( ord_le1824328871od_a_a
% 2.17/1.13 = ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 2.17/1.13 ! [T: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ T @ A4 )
% 2.17/1.13 => ( member449909584od_a_a @ T @ B5 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_202_subset__iff,axiom,
% 2.17/1.13 ( ord_less_eq_set_a
% 2.17/1.13 = ( ^ [A4: set_a,B5: set_a] :
% 2.17/1.13 ! [T: a] :
% 2.17/1.13 ( ( member_a @ T @ A4 )
% 2.17/1.13 => ( member_a @ T @ B5 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_203_equalityD2,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( A3 = B3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_204_equalityD1,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( A3 = B3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_205_subset__eq,axiom,
% 2.17/1.13 ( ord_le1824328871od_a_a
% 2.17/1.13 = ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 2.17/1.13 ! [X3: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X3 @ A4 )
% 2.17/1.13 => ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_206_subset__eq,axiom,
% 2.17/1.13 ( ord_less_eq_set_a
% 2.17/1.13 = ( ^ [A4: set_a,B5: set_a] :
% 2.17/1.13 ! [X3: a] :
% 2.17/1.13 ( ( member_a @ X3 @ A4 )
% 2.17/1.13 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_207_equalityE,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( A3 = B3 )
% 2.17/1.13 => ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_208_subsetD,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: product_prod_a_a] :
% 2.17/1.13 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 2.17/1.13 => ( ( member449909584od_a_a @ C2 @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ C2 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_209_subsetD,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a,C2: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ( member_a @ C2 @ A3 )
% 2.17/1.13 => ( member_a @ C2 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_210_in__mono,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,X2: product_prod_a_a] :
% 2.17/1.13 ( ( ord_le1824328871od_a_a @ A3 @ B3 )
% 2.17/1.13 => ( ( member449909584od_a_a @ X2 @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ X2 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_211_in__mono,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a,X2: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ( member_a @ X2 @ A3 )
% 2.17/1.13 => ( member_a @ X2 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_212_Pair__mono,axiom,
% 2.17/1.13 ! [X2: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X2 @ X5 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 2.17/1.13 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_213_bot__prod__def,axiom,
% 2.17/1.13 ( bot_bo1519632275_set_a
% 2.17/1.13 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_214_order__refl,axiom,
% 2.17/1.13 ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_215_subset__emptyI,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [X4: product_prod_a_a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_216_subset__emptyI,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ! [X4: a] :
% 2.17/1.13 ~ ( member_a @ X4 @ A3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_217_bot_Oextremum__uniqueI,axiom,
% 2.17/1.13 ! [A: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 2.17/1.13 => ( A = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_218_bot_Oextremum__unique,axiom,
% 2.17/1.13 ! [A: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 2.17/1.13 = ( A = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_219_bot_Oextremum,axiom,
% 2.17/1.13 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_220_bot__set__def,axiom,
% 2.17/1.13 ( bot_bot_set_a
% 2.17/1.13 = ( collect_a @ bot_bot_a_o ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_221_bot__empty__eq,axiom,
% 2.17/1.13 ( bot_bo1293121322_a_a_o
% 2.17/1.13 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_222_bot__empty__eq,axiom,
% 2.17/1.13 ( bot_bot_a_o
% 2.17/1.13 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_223_order__subst1,axiom,
% 2.17/1.13 ! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ ( F @ B ) )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B @ C2 )
% 2.17/1.13 => ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_224_order__subst2,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
% 2.17/1.13 => ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_225_ord__eq__le__subst,axiom,
% 2.17/1.13 ! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
% 2.17/1.13 ( ( A
% 2.17/1.13 = ( F @ B ) )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B @ C2 )
% 2.17/1.13 => ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_226_ord__le__eq__subst,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( ( ( F @ B )
% 2.17/1.13 = C2 )
% 2.17/1.13 => ( ! [X4: set_a,Y3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_227_eq__iff,axiom,
% 2.17/1.13 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.17/1.13 = ( ^ [X3: set_a,Y2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 2.17/1.13 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_228_antisym,axiom,
% 2.17/1.13 ! [X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ Y @ X2 )
% 2.17/1.13 => ( X2 = Y ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_229_eq__refl,axiom,
% 2.17/1.13 ! [X2: set_a,Y: set_a] :
% 2.17/1.13 ( ( X2 = Y )
% 2.17/1.13 => ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_230_order_Otrans,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B @ C2 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_231_antisym__conv,axiom,
% 2.17/1.13 ! [Y: set_a,X2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ Y @ X2 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.17/1.13 = ( X2 = Y ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_232_order__class_Oorder_Oeq__iff,axiom,
% 2.17/1.13 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.17/1.13 = ( ^ [A6: set_a,B6: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A6 @ B6 )
% 2.17/1.13 & ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_233_ord__eq__le__trans,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,C2: set_a] :
% 2.17/1.13 ( ( A = B )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B @ C2 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_234_ord__le__eq__trans,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( ( B = C2 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_235_order__class_Oorder_Oantisym,axiom,
% 2.17/1.13 ! [A: set_a,B: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ B @ A )
% 2.17/1.13 => ( A = B ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_236_order__trans,axiom,
% 2.17/1.13 ! [X2: set_a,Y: set_a,Z: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X2 @ Y )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ Y @ Z )
% 2.17/1.13 => ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_237_dual__order_Orefl,axiom,
% 2.17/1.13 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_238_dual__order_Otrans,axiom,
% 2.17/1.13 ! [B: set_a,A: set_a,C2: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ B @ A )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ C2 @ B )
% 2.17/1.13 => ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_239_dual__order_Oeq__iff,axiom,
% 2.17/1.13 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 2.17/1.13 = ( ^ [A6: set_a,B6: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ B6 @ A6 )
% 2.17/1.13 & ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_240_dual__order_Oantisym,axiom,
% 2.17/1.13 ! [B: set_a,A: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ B @ A )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ A @ B )
% 2.17/1.13 => ( A = B ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_241_ssubst__Pair__rhs,axiom,
% 2.17/1.13 ! [R: a,S: a,R3: set_Product_prod_a_a,S2: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R3 )
% 2.17/1.13 => ( ( S2 = S )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_242_transitivity,axiom,
% 2.17/1.13 trans_a @ relation ).
% 2.17/1.13
% 2.17/1.13 thf(fact_243_Set_Ois__empty__def,axiom,
% 2.17/1.13 ( is_empty_a
% 2.17/1.13 = ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_244_transD,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.13 ( ( trans_a @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_245_transE,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
% 2.17/1.13 ( ( trans_a @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_246_transI,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a] :
% 2.17/1.13 ( ! [X4: a,Y3: a,Z3: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
% 2.17/1.13 => ( trans_a @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_247_trans__def,axiom,
% 2.17/1.13 ( trans_a
% 2.17/1.13 = ( ^ [R2: set_Product_prod_a_a] :
% 2.17/1.13 ! [X3: a,Y2: a,Z4: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_248_trans__empty,axiom,
% 2.17/1.13 trans_a @ bot_bo2131659635od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_249_preference_Otransitivity,axiom,
% 2.17/1.13 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.17/1.13 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.17/1.13 => ( trans_a @ Relation ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_250_preorder__on__def,axiom,
% 2.17/1.13 ( order_preorder_on_a
% 2.17/1.13 = ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
% 2.17/1.13 ( ( refl_on_a @ A4 @ R2 )
% 2.17/1.13 & ( trans_a @ R2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_251_Collect__empty__eq__bot,axiom,
% 2.17/1.13 ! [P: a > $o] :
% 2.17/1.13 ( ( ( collect_a @ P )
% 2.17/1.13 = bot_bot_set_a )
% 2.17/1.13 = ( P = bot_bot_a_o ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_252_relChain__def,axiom,
% 2.17/1.13 ( bNF_Ca850758276_set_a
% 2.17/1.13 = ( ^ [R2: set_Product_prod_a_a,As: a > set_a] :
% 2.17/1.13 ! [I: a,J: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_253_Id__on__empty,axiom,
% 2.17/1.13 ( ( id_on_a @ bot_bot_set_a )
% 2.17/1.13 = bot_bo2131659635od_a_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_254_Id__onI,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_255_Id__onI,axiom,
% 2.17/1.13 ! [A: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ A @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_256_Id__on__iff,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,Y: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ ( id_on_1182990020od_a_a @ A3 ) )
% 2.17/1.13 = ( ( X2 = Y )
% 2.17/1.13 & ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_257_Id__on__iff,axiom,
% 2.17/1.13 ! [X2: a,Y: a,A3: set_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( id_on_a @ A3 ) )
% 2.17/1.13 = ( ( X2 = Y )
% 2.17/1.13 & ( member_a @ X2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_258_Id__on__eqI,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( A = B )
% 2.17/1.13 => ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_259_Id__on__eqI,axiom,
% 2.17/1.13 ! [A: a,B: a,A3: set_a] :
% 2.17/1.13 ( ( A = B )
% 2.17/1.13 => ( ( member_a @ A @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_260_Id__onE,axiom,
% 2.17/1.13 ! [C2: produc1572603623od_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member2057358096od_a_a @ C2 @ ( id_on_1182990020od_a_a @ A3 ) )
% 2.17/1.13 => ~ ! [X4: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( C2
% 2.17/1.13 != ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_261_Id__onE,axiom,
% 2.17/1.13 ! [C2: product_prod_a_a,A3: set_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ C2 @ ( id_on_a @ A3 ) )
% 2.17/1.13 => ~ ! [X4: a] :
% 2.17/1.13 ( ( member_a @ X4 @ A3 )
% 2.17/1.13 => ( C2
% 2.17/1.13 != ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_262_trans__Id__on,axiom,
% 2.17/1.13 ! [A3: set_a] : ( trans_a @ ( id_on_a @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_263_refl__on__Id__on,axiom,
% 2.17/1.13 ! [A3: set_a] : ( refl_on_a @ A3 @ ( id_on_a @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_264_under__incr,axiom,
% 2.17/1.13 ! [R: set_Product_prod_a_a,A: a,B: a] :
% 2.17/1.13 ( ( trans_a @ R )
% 2.17/1.13 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( order_under_a @ R @ A ) @ ( order_under_a @ R @ B ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_265_GreatestI2__order,axiom,
% 2.17/1.13 ! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
% 2.17/1.13 ( ( P @ X2 )
% 2.17/1.13 => ( ! [Y3: set_a] :
% 2.17/1.13 ( ( P @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ Y3 @ X2 ) )
% 2.17/1.13 => ( ! [X4: set_a] :
% 2.17/1.13 ( ( P @ X4 )
% 2.17/1.13 => ( ! [Y6: set_a] :
% 2.17/1.13 ( ( P @ Y6 )
% 2.17/1.13 => ( ord_less_eq_set_a @ Y6 @ X4 ) )
% 2.17/1.13 => ( Q @ X4 ) ) )
% 2.17/1.13 => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_266_Greatest__equality,axiom,
% 2.17/1.13 ! [P: set_a > $o,X2: set_a] :
% 2.17/1.13 ( ( P @ X2 )
% 2.17/1.13 => ( ! [Y3: set_a] :
% 2.17/1.13 ( ( P @ Y3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ Y3 @ X2 ) )
% 2.17/1.13 => ( ( order_Greatest_set_a @ P )
% 2.17/1.13 = X2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_267_trans__singleton,axiom,
% 2.17/1.13 ! [A: a] : ( trans_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ A ) @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_268_insert__iff,axiom,
% 2.17/1.13 ! [A: a,B: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 2.17/1.13 = ( ( A = B )
% 2.17/1.13 | ( member_a @ A @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_269_insert__iff,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 2.17/1.13 = ( ( A = B )
% 2.17/1.13 | ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_270_insertCI,axiom,
% 2.17/1.13 ! [A: a,B3: set_a,B: a] :
% 2.17/1.13 ( ( ~ ( member_a @ A @ B3 )
% 2.17/1.13 => ( A = B ) )
% 2.17/1.13 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_271_insertCI,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
% 2.17/1.13 ( ( ~ ( member449909584od_a_a @ A @ B3 )
% 2.17/1.13 => ( A = B ) )
% 2.17/1.13 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_272_singletonI,axiom,
% 2.17/1.13 ! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_273_singletonI,axiom,
% 2.17/1.13 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_274_insert__subset,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ A3 ) @ B3 )
% 2.17/1.13 = ( ( member449909584od_a_a @ X2 @ B3 )
% 2.17/1.13 & ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_275_insert__subset,axiom,
% 2.17/1.13 ! [X2: a,A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
% 2.17/1.13 = ( ( member_a @ X2 @ B3 )
% 2.17/1.13 & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_276_singleton__insert__inj__eq,axiom,
% 2.17/1.13 ! [B: a,A: a,A3: set_a] :
% 2.17/1.13 ( ( ( insert_a @ B @ bot_bot_set_a )
% 2.17/1.13 = ( insert_a @ A @ A3 ) )
% 2.17/1.13 = ( ( A = B )
% 2.17/1.13 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_277_singleton__insert__inj__eq_H,axiom,
% 2.17/1.13 ! [A: a,A3: set_a,B: a] :
% 2.17/1.13 ( ( ( insert_a @ A @ A3 )
% 2.17/1.13 = ( insert_a @ B @ bot_bot_set_a ) )
% 2.17/1.13 = ( ( A = B )
% 2.17/1.13 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_278_prod_Ocollapse,axiom,
% 2.17/1.13 ! [Prod: product_prod_a_a] :
% 2.17/1.13 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 2.17/1.13 = Prod ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_279_insert__subsetI,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,X6: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X2 @ A3 )
% 2.17/1.13 => ( ( ord_le1824328871od_a_a @ X6 @ A3 )
% 2.17/1.13 => ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ X6 ) @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_280_insert__subsetI,axiom,
% 2.17/1.13 ! [X2: a,A3: set_a,X6: set_a] :
% 2.17/1.13 ( ( member_a @ X2 @ A3 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ X6 @ A3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( insert_a @ X2 @ X6 ) @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_281_less__eq__prod__def,axiom,
% 2.17/1.13 ( ord_le486764743_set_a
% 2.17/1.13 = ( ^ [X3: produc1691597095_set_a,Y2: produc1691597095_set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ ( produc1926412547_set_a @ X3 ) @ ( produc1926412547_set_a @ Y2 ) )
% 2.17/1.13 & ( ord_less_eq_set_a @ ( produc918972485_set_a @ X3 ) @ ( produc918972485_set_a @ Y2 ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_282_singleton__inject,axiom,
% 2.17/1.13 ! [A: a,B: a] :
% 2.17/1.13 ( ( ( insert_a @ A @ bot_bot_set_a )
% 2.17/1.13 = ( insert_a @ B @ bot_bot_set_a ) )
% 2.17/1.13 => ( A = B ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_283_insert__not__empty,axiom,
% 2.17/1.13 ! [A: a,A3: set_a] :
% 2.17/1.13 ( ( insert_a @ A @ A3 )
% 2.17/1.13 != bot_bot_set_a ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_284_doubleton__eq__iff,axiom,
% 2.17/1.13 ! [A: a,B: a,C2: a,D: a] :
% 2.17/1.13 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 2.17/1.13 = ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 2.17/1.13 = ( ( ( A = C2 )
% 2.17/1.13 & ( B = D ) )
% 2.17/1.13 | ( ( A = D )
% 2.17/1.13 & ( B = C2 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_285_singleton__iff,axiom,
% 2.17/1.13 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 2.17/1.13 = ( B = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_286_singleton__iff,axiom,
% 2.17/1.13 ! [B: a,A: a] :
% 2.17/1.13 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 2.17/1.13 = ( B = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_287_singletonD,axiom,
% 2.17/1.13 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 2.17/1.13 => ( B = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_288_singletonD,axiom,
% 2.17/1.13 ! [B: a,A: a] :
% 2.17/1.13 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 2.17/1.13 => ( B = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_289_subset__insertI2,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a,B: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_290_subset__insertI,axiom,
% 2.17/1.13 ! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_291_subset__insert,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.17/1.13 ( ~ ( member449909584od_a_a @ X2 @ A3 )
% 2.17/1.13 => ( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X2 @ B3 ) )
% 2.17/1.13 = ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_292_subset__insert,axiom,
% 2.17/1.13 ! [X2: a,A3: set_a,B3: set_a] :
% 2.17/1.13 ( ~ ( member_a @ X2 @ A3 )
% 2.17/1.13 => ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
% 2.17/1.13 = ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_293_insert__mono,axiom,
% 2.17/1.13 ! [C3: set_a,D2: set_a,A: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ C3 @ D2 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_294_mk__disjoint__insert,axiom,
% 2.17/1.13 ! [A: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ A @ A3 )
% 2.17/1.13 => ? [B7: set_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert_a @ A @ B7 ) )
% 2.17/1.13 & ~ ( member_a @ A @ B7 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_295_mk__disjoint__insert,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ? [B7: set_Product_prod_a_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert1116662519od_a_a @ A @ B7 ) )
% 2.17/1.13 & ~ ( member449909584od_a_a @ A @ B7 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_296_insert__eq__iff,axiom,
% 2.17/1.13 ! [A: a,A3: set_a,B: a,B3: set_a] :
% 2.17/1.13 ( ~ ( member_a @ A @ A3 )
% 2.17/1.13 => ( ~ ( member_a @ B @ B3 )
% 2.17/1.13 => ( ( ( insert_a @ A @ A3 )
% 2.17/1.13 = ( insert_a @ B @ B3 ) )
% 2.17/1.13 = ( ( ( A = B )
% 2.17/1.13 => ( A3 = B3 ) )
% 2.17/1.13 & ( ( A != B )
% 2.17/1.13 => ? [C4: set_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert_a @ B @ C4 ) )
% 2.17/1.13 & ~ ( member_a @ B @ C4 )
% 2.17/1.13 & ( B3
% 2.17/1.13 = ( insert_a @ A @ C4 ) )
% 2.17/1.13 & ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_297_insert__eq__iff,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.17/1.13 ( ~ ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ( ~ ( member449909584od_a_a @ B @ B3 )
% 2.17/1.13 => ( ( ( insert1116662519od_a_a @ A @ A3 )
% 2.17/1.13 = ( insert1116662519od_a_a @ B @ B3 ) )
% 2.17/1.13 = ( ( ( A = B )
% 2.17/1.13 => ( A3 = B3 ) )
% 2.17/1.13 & ( ( A != B )
% 2.17/1.13 => ? [C4: set_Product_prod_a_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert1116662519od_a_a @ B @ C4 ) )
% 2.17/1.13 & ~ ( member449909584od_a_a @ B @ C4 )
% 2.17/1.13 & ( B3
% 2.17/1.13 = ( insert1116662519od_a_a @ A @ C4 ) )
% 2.17/1.13 & ~ ( member449909584od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_298_insert__absorb,axiom,
% 2.17/1.13 ! [A: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ A @ A3 )
% 2.17/1.13 => ( ( insert_a @ A @ A3 )
% 2.17/1.13 = A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_299_insert__absorb,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ A3 )
% 2.17/1.13 => ( ( insert1116662519od_a_a @ A @ A3 )
% 2.17/1.13 = A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_300_insert__ident,axiom,
% 2.17/1.13 ! [X2: a,A3: set_a,B3: set_a] :
% 2.17/1.13 ( ~ ( member_a @ X2 @ A3 )
% 2.17/1.13 => ( ~ ( member_a @ X2 @ B3 )
% 2.17/1.13 => ( ( ( insert_a @ X2 @ A3 )
% 2.17/1.13 = ( insert_a @ X2 @ B3 ) )
% 2.17/1.13 = ( A3 = B3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_301_insert__ident,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 2.17/1.13 ( ~ ( member449909584od_a_a @ X2 @ A3 )
% 2.17/1.13 => ( ~ ( member449909584od_a_a @ X2 @ B3 )
% 2.17/1.13 => ( ( ( insert1116662519od_a_a @ X2 @ A3 )
% 2.17/1.13 = ( insert1116662519od_a_a @ X2 @ B3 ) )
% 2.17/1.13 = ( A3 = B3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_302_Set_Oset__insert,axiom,
% 2.17/1.13 ! [X2: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ X2 @ A3 )
% 2.17/1.13 => ~ ! [B7: set_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert_a @ X2 @ B7 ) )
% 2.17/1.13 => ( member_a @ X2 @ B7 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_303_Set_Oset__insert,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X2 @ A3 )
% 2.17/1.13 => ~ ! [B7: set_Product_prod_a_a] :
% 2.17/1.13 ( ( A3
% 2.17/1.13 = ( insert1116662519od_a_a @ X2 @ B7 ) )
% 2.17/1.13 => ( member449909584od_a_a @ X2 @ B7 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_304_insertI2,axiom,
% 2.17/1.13 ! [A: a,B3: set_a,B: a] :
% 2.17/1.13 ( ( member_a @ A @ B3 )
% 2.17/1.13 => ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_305_insertI2,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ B3 )
% 2.17/1.13 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_306_insertI1,axiom,
% 2.17/1.13 ! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_307_insertI1,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B3: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_308_insertE,axiom,
% 2.17/1.13 ! [A: a,B: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 2.17/1.13 => ( ( A != B )
% 2.17/1.13 => ( member_a @ A @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_309_insertE,axiom,
% 2.17/1.13 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 2.17/1.13 => ( ( A != B )
% 2.17/1.13 => ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_310_surjective__pairing,axiom,
% 2.17/1.13 ! [T2: product_prod_a_a] :
% 2.17/1.13 ( T2
% 2.17/1.13 = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_311_prod_Oexhaust__sel,axiom,
% 2.17/1.13 ! [Prod: product_prod_a_a] :
% 2.17/1.13 ( Prod
% 2.17/1.13 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_312_fst__conv,axiom,
% 2.17/1.13 ! [X1: a,X22: a] :
% 2.17/1.13 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.17/1.13 = X1 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_313_fst__eqD,axiom,
% 2.17/1.13 ! [X2: a,Y: a,A: a] :
% 2.17/1.13 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 2.17/1.13 = A )
% 2.17/1.13 => ( X2 = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_314_snd__conv,axiom,
% 2.17/1.13 ! [X1: a,X22: a] :
% 2.17/1.13 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.17/1.13 = X22 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_315_snd__eqD,axiom,
% 2.17/1.13 ! [X2: a,Y: a,A: a] :
% 2.17/1.13 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 2.17/1.13 = A )
% 2.17/1.13 => ( Y = A ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_316_refl__on__singleton,axiom,
% 2.17/1.13 ! [X2: a] : ( refl_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_317_total__on__singleton,axiom,
% 2.17/1.13 ! [X2: a] : ( total_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_318_subset__singletonD,axiom,
% 2.17/1.13 ! [A3: set_a,X2: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) )
% 2.17/1.13 => ( ( A3 = bot_bot_set_a )
% 2.17/1.13 | ( A3
% 2.17/1.13 = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_319_subset__singleton__iff,axiom,
% 2.17/1.13 ! [X6: set_a,A: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
% 2.17/1.13 = ( ( X6 = bot_bot_set_a )
% 2.17/1.13 | ( X6
% 2.17/1.13 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_320_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.17/1.13 ! [P: a > a > $o,X2: a,Y: a,A: product_prod_a_a] :
% 2.17/1.13 ( ( P @ X2 @ Y )
% 2.17/1.13 => ( ( A
% 2.17/1.13 = ( product_Pair_a_a @ X2 @ Y ) )
% 2.17/1.13 => ( P @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_321_conjI__realizer,axiom,
% 2.17/1.13 ! [P: a > $o,P2: a,Q: a > $o,Q2: a] :
% 2.17/1.13 ( ( P @ P2 )
% 2.17/1.13 => ( ( Q @ Q2 )
% 2.17/1.13 => ( ( P @ ( product_fst_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) )
% 2.17/1.13 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_322_exI__realizer,axiom,
% 2.17/1.13 ! [P: a > a > $o,Y: a,X2: a] :
% 2.17/1.13 ( ( P @ Y @ X2 )
% 2.17/1.13 => ( P @ ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_323_the__elem__eq,axiom,
% 2.17/1.13 ! [X2: a] :
% 2.17/1.13 ( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
% 2.17/1.13 = X2 ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_324_is__singletonI,axiom,
% 2.17/1.13 ! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_325_is__singleton__the__elem,axiom,
% 2.17/1.13 ( is_singleton_a
% 2.17/1.13 = ( ^ [A4: set_a] :
% 2.17/1.13 ( A4
% 2.17/1.13 = ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_326_is__singletonI_H,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( A3 != bot_bo2131659635od_a_a )
% 2.17/1.13 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member449909584od_a_a @ Y3 @ A3 )
% 2.17/1.13 => ( X4 = Y3 ) ) )
% 2.17/1.13 => ( is_sin877026331od_a_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_327_is__singletonI_H,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( A3 != bot_bot_set_a )
% 2.17/1.13 => ( ! [X4: a,Y3: a] :
% 2.17/1.13 ( ( member_a @ X4 @ A3 )
% 2.17/1.13 => ( ( member_a @ Y3 @ A3 )
% 2.17/1.13 => ( X4 = Y3 ) ) )
% 2.17/1.13 => ( is_singleton_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_328_is__singletonE,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( is_singleton_a @ A3 )
% 2.17/1.13 => ~ ! [X4: a] :
% 2.17/1.13 ( A3
% 2.17/1.13 != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_329_is__singleton__def,axiom,
% 2.17/1.13 ( is_singleton_a
% 2.17/1.13 = ( ^ [A4: set_a] :
% 2.17/1.13 ? [X3: a] :
% 2.17/1.13 ( A4
% 2.17/1.13 = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_330_linear__order__on__singleton,axiom,
% 2.17/1.13 ! [X2: a] : ( order_653178931r_on_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ bot_bo2131659635od_a_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_331_sndI,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,Y: a,Z: a] :
% 2.17/1.13 ( ( X2
% 2.17/1.13 = ( product_Pair_a_a @ Y @ Z ) )
% 2.17/1.13 => ( ( product_snd_a_a @ X2 )
% 2.17/1.13 = Z ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_332_lnear__order__on__empty,axiom,
% 2.17/1.13 order_653178931r_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 2.17/1.13
% 2.17/1.13 thf(fact_333_fstI,axiom,
% 2.17/1.13 ! [X2: product_prod_a_a,Y: a,Z: a] :
% 2.17/1.13 ( ( X2
% 2.17/1.13 = ( product_Pair_a_a @ Y @ Z ) )
% 2.17/1.13 => ( ( product_fst_a_a @ X2 )
% 2.17/1.13 = Y ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_334_eq__snd__iff,axiom,
% 2.17/1.13 ! [B: a,P2: product_prod_a_a] :
% 2.17/1.13 ( ( B
% 2.17/1.13 = ( product_snd_a_a @ P2 ) )
% 2.17/1.13 = ( ? [A6: a] :
% 2.17/1.13 ( P2
% 2.17/1.13 = ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_335_eq__fst__iff,axiom,
% 2.17/1.13 ! [A: a,P2: product_prod_a_a] :
% 2.17/1.13 ( ( A
% 2.17/1.13 = ( product_fst_a_a @ P2 ) )
% 2.17/1.13 = ( ? [B6: a] :
% 2.17/1.13 ( P2
% 2.17/1.13 = ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_336_prod_Oswap__def,axiom,
% 2.17/1.13 ( product_swap_a_a
% 2.17/1.13 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_337_Range__insert,axiom,
% 2.17/1.13 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( range_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
% 2.17/1.13 = ( insert_a @ B @ ( range_a_a @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_338_swap__simp,axiom,
% 2.17/1.13 ! [X2: a,Y: a] :
% 2.17/1.13 ( ( product_swap_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
% 2.17/1.13 = ( product_Pair_a_a @ Y @ X2 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_339_RangeE,axiom,
% 2.17/1.13 ! [B: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ B @ ( range_a_a @ R ) )
% 2.17/1.13 => ~ ! [A5: a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B ) @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_340_Range__iff,axiom,
% 2.17/1.13 ! [A: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ A @ ( range_a_a @ R ) )
% 2.17/1.13 = ( ? [Y2: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ A ) @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_341_Range_Ocases,axiom,
% 2.17/1.13 ! [A: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ A @ ( range_a_a @ R ) )
% 2.17/1.13 => ~ ! [A5: a] :
% 2.17/1.13 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ A ) @ R ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_342_Range_Osimps,axiom,
% 2.17/1.13 ! [A: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member_a @ A @ ( range_a_a @ R ) )
% 2.17/1.13 = ( ? [A6: a,B6: a] :
% 2.17/1.13 ( ( A = B6 )
% 2.17/1.13 & ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ R ) ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_343_Range_Ointros,axiom,
% 2.17/1.13 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 2.17/1.13 => ( member_a @ B @ ( range_a_a @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_344_Range_Oinducts,axiom,
% 2.17/1.13 ! [X2: a,R: set_Product_prod_a_a,P: a > $o] :
% 2.17/1.13 ( ( member_a @ X2 @ ( range_a_a @ R ) )
% 2.17/1.13 => ( ! [A5: a,B4: a] :
% 2.17/1.13 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B4 ) @ R )
% 2.17/1.13 => ( P @ B4 ) )
% 2.17/1.13 => ( P @ X2 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_345_Domain__insert,axiom,
% 2.17/1.13 ! [A: a,B: a,R: set_Product_prod_a_a] :
% 2.17/1.13 ( ( domain_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
% 2.17/1.13 = ( insert_a @ A @ ( domain_a_a @ R ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_346_subset__Compl__singleton,axiom,
% 2.17/1.13 ! [A3: set_Product_prod_a_a,B: product_prod_a_a] :
% 2.17/1.13 ( ( ord_le1824328871od_a_a @ A3 @ ( uminus1440301054od_a_a @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) )
% 2.17/1.13 = ( ~ ( member449909584od_a_a @ B @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_347_subset__Compl__singleton,axiom,
% 2.17/1.13 ! [A3: set_a,B: a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
% 2.17/1.13 = ( ~ ( member_a @ B @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_348_Compl__iff,axiom,
% 2.17/1.13 ! [C2: a,A3: set_a] :
% 2.17/1.13 ( ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) )
% 2.17/1.13 = ( ~ ( member_a @ C2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_349_Compl__iff,axiom,
% 2.17/1.13 ! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) )
% 2.17/1.13 = ( ~ ( member449909584od_a_a @ C2 @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_350_ComplI,axiom,
% 2.17/1.13 ! [C2: a,A3: set_a] :
% 2.17/1.13 ( ~ ( member_a @ C2 @ A3 )
% 2.17/1.13 => ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_351_ComplI,axiom,
% 2.17/1.13 ! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
% 2.17/1.13 ( ~ ( member449909584od_a_a @ C2 @ A3 )
% 2.17/1.13 => ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_352_Compl__anti__mono,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 2.17/1.13 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_353_Compl__subset__Compl__iff,axiom,
% 2.17/1.13 ! [A3: set_a,B3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B3 ) )
% 2.17/1.13 = ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(fact_354_subset__Compl__self__eq,axiom,
% 2.17/1.13 ! [A3: set_a] :
% 2.17/1.13 ( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
% 2.17/1.13 = ( A3 = bot_bot_set_a ) ) ).
% 2.17/1.13
% 2.17/1.13 thf(conj_0,conjecture,
% 2.17/1.13 ( ( member_a @ z @ carrier )
% 2.17/1.13 & ( member_a @ y @ carrier ) ) ).
% 2.17/1.13 ------------------------
% 2.17/1.15 % (7499)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/2Mi)
% 2.17/1.15 % (7497)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/4Mi)
% 2.17/1.15 % (7498)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/27Mi)
% 2.17/1.15 % (7496)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/183Mi)
% 2.17/1.15 % (7500)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/2Mi)
% 2.17/1.15 % (7501)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/275Mi)
% 2.17/1.15 % (7502)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/18Mi)
% 2.17/1.15 % (7499)Instruction limit reached!
% 2.17/1.15 % (7499)------------------------------
% 2.17/1.15 % (7499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.15 % (7499)Termination reason: Unknown
% 2.17/1.15 % (7499)Termination phase: shuffling
% 2.17/1.15
% 2.17/1.15 % (7499)Memory used [KB]: 1407
% 2.17/1.15 % (7499)Time elapsed: 0.002 s
% 2.17/1.15 % (7499)Instructions burned: 2 (million)
% 2.17/1.15 % (7499)------------------------------
% 2.17/1.15 % (7499)------------------------------
% 2.17/1.15 % (7500)Instruction limit reached!
% 2.17/1.15 % (7500)------------------------------
% 2.17/1.15 % (7500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.15 % (7500)Termination reason: Unknown
% 2.17/1.15 % (7500)Termination phase: shuffling
% 2.17/1.15
% 2.17/1.15 % (7500)Memory used [KB]: 1407
% 2.17/1.15 % (7500)Time elapsed: 0.002 s
% 2.17/1.15 % (7500)Instructions burned: 2 (million)
% 2.17/1.15 % (7500)------------------------------
% 2.17/1.15 % (7500)------------------------------
% 2.17/1.15 % (7497)Instruction limit reached!
% 2.17/1.15 % (7497)------------------------------
% 2.17/1.15 % (7497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.15 % (7497)Termination reason: Unknown
% 2.17/1.15 % (7497)Termination phase: shuffling
% 2.17/1.15
% 2.17/1.15 % (7497)Memory used [KB]: 1535
% 2.17/1.15 % (7497)Time elapsed: 0.003 s
% 2.17/1.15 % (7497)Instructions burned: 6 (million)
% 2.17/1.15 % (7497)------------------------------
% 2.17/1.15 % (7497)------------------------------
% 2.17/1.16 % (7502)Instruction limit reached!
% 2.17/1.16 % (7502)------------------------------
% 2.17/1.16 % (7502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.16 % (7502)Termination reason: Unknown
% 2.17/1.16 % (7502)Termination phase: shuffling
% 2.17/1.16
% 2.17/1.16 % (7502)Memory used [KB]: 1791
% 2.17/1.16 % (7502)Time elapsed: 0.007 s
% 2.17/1.16 % (7502)Instructions burned: 18 (million)
% 2.17/1.16 % (7502)------------------------------
% 2.17/1.16 % (7502)------------------------------
% 2.17/1.16 % (7498)Instruction limit reached!
% 2.17/1.16 % (7498)------------------------------
% 2.17/1.16 % (7498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.16 % (7498)Termination reason: Unknown
% 2.17/1.16 % (7498)Termination phase: shuffling
% 2.17/1.16
% 2.17/1.16 % (7498)Memory used [KB]: 1918
% 2.17/1.16 % (7498)Time elapsed: 0.010 s
% 2.17/1.16 % (7498)Instructions burned: 28 (million)
% 2.17/1.16 % (7498)------------------------------
% 2.17/1.16 % (7498)------------------------------
% 2.17/1.16 % (7503)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/3Mi)
% 2.17/1.16 % (7504)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/37Mi)
% 2.17/1.16 % (7505)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/15Mi)
% 2.17/1.16 % (7503)Instruction limit reached!
% 2.17/1.16 % (7503)------------------------------
% 2.17/1.16 % (7503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.16 % (7503)Termination reason: Unknown
% 2.17/1.16 % (7503)Termination phase: shuffling
% 2.17/1.16
% 2.17/1.16 % (7503)Memory used [KB]: 1407
% 2.17/1.16 % (7503)Time elapsed: 0.002 s
% 2.17/1.16 % (7503)Instructions burned: 3 (million)
% 2.17/1.16 % (7503)------------------------------
% 2.17/1.16 % (7503)------------------------------
% 2.17/1.16 % (7506)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/3Mi)
% 2.17/1.17 % (7505)Instruction limit reached!
% 2.17/1.17 % (7505)------------------------------
% 2.17/1.17 % (7505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.17 % (7505)Termination reason: Unknown
% 2.17/1.17 % (7505)Termination phase: shuffling
% 2.17/1.17
% 2.17/1.17 % (7505)Memory used [KB]: 1663
% 2.17/1.17 % (7505)Time elapsed: 0.006 s
% 2.17/1.17 % (7505)Instructions burned: 16 (million)
% 2.17/1.17 % (7505)------------------------------
% 2.17/1.17 % (7505)------------------------------
% 2.17/1.17 % (7506)Instruction limit reached!
% 2.17/1.17 % (7506)------------------------------
% 2.17/1.17 % (7506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.17 % (7506)Termination reason: Unknown
% 2.17/1.17 % (7506)Termination phase: shuffling
% 2.17/1.17
% 2.17/1.17 % (7506)Memory used [KB]: 1407
% 2.17/1.17 % (7506)Time elapsed: 0.002 s
% 2.17/1.17 % (7506)Instructions burned: 3 (million)
% 2.17/1.17 % (7506)------------------------------
% 2.17/1.17 % (7506)------------------------------
% 2.17/1.17 % (7507)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/1041Mi)
% 2.17/1.17 % (7501)Refutation not found, incomplete strategy
% 2.17/1.17 % (7501)------------------------------
% 2.17/1.17 % (7501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.17 % (7501)Termination reason: Refutation not found, incomplete strategy
% 2.17/1.17
% 2.17/1.17
% 2.17/1.17 % (7501)Memory used [KB]: 6524
% 2.17/1.17 % (7501)Time elapsed: 0.020 s
% 2.17/1.17 % (7501)Instructions burned: 64 (million)
% 2.17/1.17 % (7501)------------------------------
% 2.17/1.17 % (7501)------------------------------
% 2.17/1.17 % (7504)Instruction limit reached!
% 2.17/1.17 % (7504)------------------------------
% 2.17/1.17 % (7504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.17 % (7504)Termination reason: Unknown
% 2.17/1.17 % (7504)Termination phase: shuffling
% 2.17/1.17
% 2.17/1.17 % (7504)Memory used [KB]: 2174
% 2.17/1.17 % (7504)Time elapsed: 0.011 s
% 2.17/1.17 % (7504)Instructions burned: 37 (million)
% 2.17/1.17 % (7504)------------------------------
% 2.17/1.17 % (7504)------------------------------
% 2.17/1.17 % (7508)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/7Mi)
% 2.17/1.17 % (7508)Instruction limit reached!
% 2.17/1.17 % (7508)------------------------------
% 2.17/1.17 % (7508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.17 % (7508)Termination reason: Unknown
% 2.17/1.17 % (7508)Termination phase: shuffling
% 2.17/1.17
% 2.17/1.17 % (7508)Memory used [KB]: 1535
% 2.17/1.17 % (7508)Time elapsed: 0.004 s
% 2.17/1.17 % (7508)Instructions burned: 9 (million)
% 2.17/1.17 % (7508)------------------------------
% 2.17/1.17 % (7508)------------------------------
% 2.17/1.17 % (7509)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/16Mi)
% 2.17/1.17 % (7510)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/3Mi)
% 2.17/1.18 % (7510)Instruction limit reached!
% 2.17/1.18 % (7510)------------------------------
% 2.17/1.18 % (7510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.18 % (7510)Termination reason: Unknown
% 2.17/1.18 % (7510)Termination phase: shuffling
% 2.17/1.18
% 2.17/1.18 % (7510)Memory used [KB]: 1535
% 2.17/1.18 % (7510)Time elapsed: 0.003 s
% 2.17/1.18 % (7510)Instructions burned: 5 (million)
% 2.17/1.18 % (7510)------------------------------
% 2.17/1.18 % (7510)------------------------------
% 2.17/1.18 % (7511)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/3Mi)
% 2.17/1.18 % (7512)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/7Mi)
% 2.17/1.18 % (7509)Instruction limit reached!
% 2.17/1.18 % (7509)------------------------------
% 2.17/1.18 % (7509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.18 % (7509)Termination reason: Unknown
% 2.17/1.18 % (7509)Termination phase: shuffling
% 2.17/1.18
% 2.17/1.18 % (7509)Memory used [KB]: 1791
% 2.17/1.18 % (7509)Time elapsed: 0.007 s
% 2.17/1.18 % (7509)Instructions burned: 18 (million)
% 2.17/1.18 % (7509)------------------------------
% 2.17/1.18 % (7509)------------------------------
% 2.17/1.18 % (7511)Instruction limit reached!
% 2.17/1.18 % (7511)------------------------------
% 2.17/1.18 % (7511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.18 % (7511)Termination reason: Unknown
% 2.17/1.18 % (7511)Termination phase: shuffling
% 2.17/1.18
% 2.17/1.18 % (7511)Memory used [KB]: 1535
% 2.17/1.18 % (7511)Time elapsed: 0.003 s
% 2.17/1.18 % (7511)Instructions burned: 5 (million)
% 2.17/1.18 % (7511)------------------------------
% 2.17/1.18 % (7511)------------------------------
% 2.17/1.18 % (7512)Instruction limit reached!
% 2.17/1.18 % (7512)------------------------------
% 2.17/1.18 % (7512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.18 % (7512)Termination reason: Unknown
% 2.17/1.18 % (7512)Termination phase: shuffling
% 2.17/1.18
% 2.17/1.18 % (7512)Memory used [KB]: 1535
% 2.17/1.18 % (7512)Time elapsed: 0.004 s
% 2.17/1.18 % (7512)Instructions burned: 8 (million)
% 2.17/1.18 % (7512)------------------------------
% 2.17/1.18 % (7512)------------------------------
% 2.17/1.18 % (7513)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/3Mi)
% 2.17/1.18 % (7513)Instruction limit reached!
% 2.17/1.18 % (7513)------------------------------
% 2.17/1.18 % (7513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.18 % (7513)Termination reason: Unknown
% 2.17/1.18 % (7513)Termination phase: shuffling
% 2.17/1.18
% 2.17/1.18 % (7513)Memory used [KB]: 1407
% 2.17/1.18 % (7513)Time elapsed: 0.002 s
% 2.17/1.18 % (7513)Instructions burned: 3 (million)
% 2.17/1.18 % (7513)------------------------------
% 2.17/1.18 % (7513)------------------------------
% 2.17/1.19 % (7514)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/4Mi)
% 2.17/1.19 % (7514)Instruction limit reached!
% 2.17/1.19 % (7514)------------------------------
% 2.17/1.19 % (7514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.19 % (7514)Termination reason: Unknown
% 2.17/1.19 % (7514)Termination phase: shuffling
% 2.17/1.19
% 2.17/1.19 % (7514)Memory used [KB]: 1535
% 2.17/1.19 % (7514)Time elapsed: 0.003 s
% 2.17/1.19 % (7514)Instructions burned: 6 (million)
% 2.17/1.19 % (7514)------------------------------
% 2.17/1.19 % (7514)------------------------------
% 2.17/1.19 % (7516)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/18Mi)
% 2.17/1.19 % (7517)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/710Mi)
% 2.17/1.19 % (7518)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/6Mi)
% 2.17/1.19 % (7519)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/902Mi)
% 2.17/1.19 % (7518)Instruction limit reached!
% 2.17/1.19 % (7518)------------------------------
% 2.17/1.19 % (7518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.19 % (7518)Termination reason: Unknown
% 2.17/1.19 % (7518)Termination phase: shuffling
% 2.17/1.19
% 2.17/1.19 % (7518)Memory used [KB]: 1535
% 2.17/1.19 % (7518)Time elapsed: 0.003 s
% 2.17/1.19 % (7518)Instructions burned: 6 (million)
% 2.17/1.19 % (7518)------------------------------
% 2.17/1.19 % (7518)------------------------------
% 2.20/1.20 % (7516)Instruction limit reached!
% 2.20/1.20 % (7516)------------------------------
% 2.20/1.20 % (7516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.20 % (7516)Termination reason: Unknown
% 2.20/1.20 % (7516)Termination phase: shuffling
% 2.20/1.20
% 2.20/1.20 % (7516)Memory used [KB]: 1791
% 2.20/1.20 % (7516)Time elapsed: 0.007 s
% 2.20/1.20 % (7516)Instructions burned: 19 (million)
% 2.20/1.20 % (7516)------------------------------
% 2.20/1.20 % (7516)------------------------------
% 2.20/1.20 % (7520)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/21Mi)
% 2.20/1.20 % (7496)Instruction limit reached!
% 2.20/1.20 % (7496)------------------------------
% 2.20/1.20 % (7496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.20 % (7496)Termination reason: Unknown
% 2.20/1.20 % (7496)Termination phase: Saturation
% 2.20/1.20
% 2.20/1.20 % (7496)Memory used [KB]: 7291
% 2.20/1.20 % (7496)Time elapsed: 0.051 s
% 2.20/1.20 % (7496)Instructions burned: 185 (million)
% 2.20/1.20 % (7496)------------------------------
% 2.20/1.20 % (7496)------------------------------
% 2.20/1.20 % (7521)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/5Mi)
% 2.20/1.20 % (7520)Instruction limit reached!
% 2.20/1.20 % (7520)------------------------------
% 2.20/1.20 % (7520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.20 % (7520)Termination reason: Unknown
% 2.20/1.20 % (7520)Termination phase: shuffling
% 2.20/1.20
% 2.20/1.20 % (7520)Memory used [KB]: 1791
% 2.20/1.20 % (7520)Time elapsed: 0.008 s
% 2.20/1.20 % (7520)Instructions burned: 22 (million)
% 2.20/1.20 % (7520)------------------------------
% 2.20/1.20 % (7520)------------------------------
% 2.20/1.20 % (7521)Instruction limit reached!
% 2.20/1.20 % (7521)------------------------------
% 2.20/1.20 % (7521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.20 % (7521)Termination reason: Unknown
% 2.20/1.20 % (7521)Termination phase: shuffling
% 2.20/1.20
% 2.20/1.20 % (7521)Memory used [KB]: 1535
% 2.20/1.20 % (7521)Time elapsed: 0.003 s
% 2.20/1.20 % (7521)Instructions burned: 6 (million)
% 2.20/1.20 % (7521)------------------------------
% 2.20/1.20 % (7521)------------------------------
% 2.20/1.21 % (7522)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/6Mi)
% 2.20/1.21 % (7522)Instruction limit reached!
% 2.20/1.21 % (7522)------------------------------
% 2.20/1.21 % (7522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.21 % (7522)Termination reason: Unknown
% 2.20/1.21 % (7522)Termination phase: shuffling
% 2.20/1.21
% 2.20/1.21 % (7522)Memory used [KB]: 1535
% 2.20/1.21 % (7522)Time elapsed: 0.003 s
% 2.20/1.21 % (7522)Instructions burned: 6 (million)
% 2.20/1.21 % (7522)------------------------------
% 2.20/1.21 % (7522)------------------------------
% 2.20/1.21 % (7523)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_7300 for (2999ds/377Mi)
% 2.20/1.21 % (7517)First to succeed.
% 2.20/1.21 % (7517)Refutation found. Thanks to Tanya!
% 2.20/1.21 % SZS status Theorem for DTF2THF_7300
% 2.20/1.21 % SZS output start Proof for DTF2THF_7300
% 2.20/1.21 thf(type_def_5, type, set_Product_prod_a_a: $tType).
% 2.20/1.21 thf(type_def_6, type, a: $tType).
% 2.20/1.21 thf(type_def_7, type, set_a: $tType).
% 2.20/1.21 thf(type_def_9, type, set_Pr1948701895od_a_a: $tType).
% 2.20/1.21 thf(type_def_10, type, set_Pr1295299783od_a_a: $tType).
% 2.20/1.21 thf(type_def_11, type, set_set_a: $tType).
% 2.20/1.21 thf(type_def_12, type, set_Pr2045688071_set_a: $tType).
% 2.20/1.21 thf(type_def_13, type, product_prod_a_a: $tType).
% 2.20/1.21 thf(type_def_14, type, produc1691597095_set_a: $tType).
% 2.20/1.21 thf(type_def_15, type, produc1572603623od_a_a: $tType).
% 2.20/1.21 thf(type_def_16, type, produc1990712039od_a_a: $tType).
% 2.20/1.21 thf(func_def_0, type, set_Pr1295299783od_a_a: $tType).
% 2.20/1.21 thf(func_def_1, type, produc1990712039od_a_a: $tType).
% 2.20/1.21 thf(func_def_2, type, set_Pr1948701895od_a_a: $tType).
% 2.20/1.21 thf(func_def_3, type, produc1572603623od_a_a: $tType).
% 2.20/1.21 thf(func_def_4, type, set_Pr2045688071_set_a: $tType).
% 2.20/1.21 thf(func_def_5, type, produc1691597095_set_a: $tType).
% 2.20/1.21 thf(func_def_6, type, set_Product_prod_a_a: $tType).
% 2.20/1.21 thf(func_def_7, type, product_prod_a_a: $tType).
% 2.20/1.21 thf(func_def_8, type, set_set_a: $tType).
% 2.20/1.21 thf(func_def_9, type, set_a: $tType).
% 2.20/1.21 thf(func_def_10, type, a: $tType).
% 2.20/1.21 thf(func_def_11, type, bNF_Ca850758276_set_a: set_Product_prod_a_a > (a > set_a) > $o).
% 2.20/1.21 thf(func_def_12, type, uminus1440301054od_a_a: set_Product_prod_a_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_13, type, uminus_uminus_set_a: set_a > set_a).
% 2.20/1.21 thf(func_def_14, type, order_653178931r_on_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_15, type, order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_16, type, order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_17, type, order_816746613_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_18, type, order_preorder_on_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_19, type, order_under_a: set_Product_prod_a_a > a > set_a).
% 2.20/1.21 thf(func_def_20, type, bot_bo1293121322_a_a_o: product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_21, type, bot_bot_a_o: a > $o).
% 2.20/1.21 thf(func_def_22, type, bot_bo1519632275_set_a: produc1691597095_set_a).
% 2.20/1.21 thf(func_def_23, type, bot_bo1116205875od_a_a: set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_24, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_25, type, bot_bot_set_set_a: set_set_a).
% 2.20/1.21 thf(func_def_26, type, bot_bot_set_a: set_a).
% 2.20/1.21 thf(func_def_27, type, ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o).
% 2.20/1.21 thf(func_def_28, type, ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_29, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_30, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 2.20/1.21 thf(func_def_31, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.20/1.21 thf(func_def_32, type, order_Greatest_set_a: (set_a > $o) > set_a).
% 2.20/1.21 thf(func_def_33, type, prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_34, type, prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_35, type, prefer270414042_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.20/1.21 thf(func_def_36, type, prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.20/1.21 thf(func_def_37, type, prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_38, type, prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_39, type, prefer1203288218_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.20/1.21 thf(func_def_40, type, prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.20/1.21 thf(func_def_41, type, prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_42, type, prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_43, type, prefer1671525929_set_a: set_a > set_set_a > set_Pr2045688071_set_a > set_set_a).
% 2.20/1.21 thf(func_def_44, type, prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a).
% 2.20/1.21 thf(func_def_45, type, prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_46, type, prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_47, type, prefer1710114912_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_48, type, prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_49, type, prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_50, type, prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_51, type, prefer484977842_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_52, type, prefer719835986ence_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_53, type, prefer1064779160od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_54, type, prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_55, type, produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a).
% 2.20/1.21 thf(func_def_56, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 2.20/1.21 thf(func_def_57, type, produc1928581911_set_a: set_a > set_a > produc1691597095_set_a).
% 2.20/1.21 thf(func_def_58, type, product_Pair_a_a: a > a > product_prod_a_a).
% 2.20/1.21 thf(func_def_59, type, produc1926412547_set_a: produc1691597095_set_a > set_a).
% 2.20/1.21 thf(func_def_60, type, product_fst_a_a: product_prod_a_a > a).
% 2.20/1.21 thf(func_def_61, type, produc918972485_set_a: produc1691597095_set_a > set_a).
% 2.20/1.21 thf(func_def_62, type, product_snd_a_a: product_prod_a_a > a).
% 2.20/1.21 thf(func_def_63, type, product_swap_a_a: product_prod_a_a > product_prod_a_a).
% 2.20/1.21 thf(func_def_64, type, domain_a_a: set_Product_prod_a_a > set_a).
% 2.20/1.21 thf(func_def_65, type, id_on_1182990020od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_66, type, id_on_a: set_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_67, type, range_a_a: set_Product_prod_a_a > set_a).
% 2.20/1.21 thf(func_def_68, type, refl_o198693862od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_69, type, refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_70, type, refl_on_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_71, type, refl_on_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_72, type, total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_73, type, total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_74, type, total_on_set_a: set_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_75, type, total_on_a: set_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_76, type, trans_a: set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_77, type, collec1635618130od_a_a: (produc1572603623od_a_a > $o) > set_Pr1948701895od_a_a).
% 2.20/1.21 thf(func_def_78, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_79, type, collect_a: (a > $o) > set_a).
% 2.20/1.21 thf(func_def_80, type, insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_81, type, insert_a: a > set_a > set_a).
% 2.20/1.21 thf(func_def_82, type, is_empty_a: set_a > $o).
% 2.20/1.21 thf(func_def_83, type, is_sin877026331od_a_a: set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_84, type, is_singleton_a: set_a > $o).
% 2.20/1.21 thf(func_def_85, type, the_elem_a: set_a > a).
% 2.20/1.21 thf(func_def_86, type, member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o).
% 2.20/1.21 thf(func_def_87, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 2.20/1.21 thf(func_def_88, type, member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o).
% 2.20/1.21 thf(func_def_89, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 2.20/1.21 thf(func_def_90, type, member_set_a: set_a > set_set_a > $o).
% 2.20/1.21 thf(func_def_91, type, member_a: a > set_a > $o).
% 2.20/1.21 thf(func_def_92, type, carrier: set_a).
% 2.20/1.21 thf(func_def_93, type, relation: set_Product_prod_a_a).
% 2.20/1.21 thf(func_def_94, type, y: a).
% 2.20/1.21 thf(func_def_95, type, z: a).
% 2.20/1.21 thf(f1049,plain,(
% 2.20/1.21 $false),
% 2.20/1.21 inference(avatar_sat_refutation,[status(thm)],[f1041,f1047,f1048])).
% 2.20/1.21 thf(f1048,plain,(
% 2.20/1.21 spl0_2),
% 2.20/1.21 inference(avatar_split_clause,[status(thm)],[f1044,f1038])).
% 2.20/1.21 thf(f1038,definition,(
% 2.20/1.21 spl0_2 <=> ((member_a @ z @ carrier) = $true)),
% 2.20/1.21 introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition])).
% 2.20/1.21 thf(f1044,plain,(
% 2.20/1.21 ((member_a @ z @ carrier) = $true)),
% 2.20/1.21 inference(trivial_inequality_removal,[status(thm)],[f1043])).
% 2.20/1.21 thf(f1043,plain,(
% 2.20/1.21 ($true != $true) | ((member_a @ z @ carrier) = $true)),
% 2.20/1.21 inference(superposition,[status(thm)],[f1029,f1018])).
% 2.20/1.21 thf(f1018,plain,(
% 2.20/1.21 ((member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ relation) = $true)),
% 2.20/1.21 inference(cnf_transformation,[status(thm)],[f826])).
% 2.20/1.21 thf(f826,plain,(
% 2.20/1.21 ((member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ relation) = $true) & ((member449909584od_a_a @ (product_Pair_a_a @ y @ z) @ relation) = $true)),
% 2.20/1.21 inference(fool_elimination,[status(thm)],[f825])).
% 2.20/1.21 thf(f825,plain,(
% 2.20/1.21 (member449909584od_a_a @ (product_Pair_a_a @ y @ z) @ relation) & (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ relation)),
% 2.20/1.21 inference(rectify,[status(thm)],[f1])).
% 2.20/1.21 thf(f1,axiom,(
% 2.20/1.21 (member449909584od_a_a @ (product_Pair_a_a @ y @ z) @ relation) & (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ relation)),
% 2.20/1.21 file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',fact_0_assms)).
% 2.20/1.21 thf(f1029,plain,(
% 2.20/1.21 ( ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) != $true) | ((member_a @ X1 @ carrier) = $true)) )),
% 2.20/1.21 inference(cnf_transformation,[status(thm)],[f1010])).
% 2.20/1.21 thf(f1010,plain,(
% 2.20/1.21 ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) != $true) | ((member_a @ X1 @ carrier) = $true))),
% 2.20/1.21 inference(ennf_transformation,[status(thm)],[f832])).
% 2.20/1.21 thf(f832,plain,(
% 2.20/1.21 ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true) => ((member_a @ X1 @ carrier) = $true))),
% 2.20/1.21 inference(fool_elimination,[status(thm)],[f831])).
% 2.20/1.21 thf(f831,plain,(
% 2.20/1.21 ! [X0 : a,X1 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) => (member_a @ X1 @ carrier))),
% 2.20/1.21 inference(rectify,[status(thm)],[f9])).
% 2.20/1.21 thf(f9,axiom,(
% 2.20/1.21 ! [X3 : a,X2 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X2 @ X3) @ relation) => (member_a @ X2 @ carrier))),
% 2.20/1.21 file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',fact_8_not__outside)).
% 2.20/1.21 thf(f1047,plain,(
% 2.20/1.21 spl0_1),
% 2.20/1.21 inference(avatar_contradiction_clause,[status(thm)],[f1046])).
% 2.20/1.21 thf(f1046,plain,(
% 2.20/1.21 $false | spl0_1),
% 2.20/1.21 inference(subsumption_resolution,[status(thm)],[f1045,f1036])).
% 2.20/1.21 thf(f1036,plain,(
% 2.20/1.21 ((member_a @ y @ carrier) != $true) | spl0_1),
% 2.20/1.21 inference(avatar_component_clause,[status(thm)],[f1034])).
% 2.20/1.21 thf(f1034,definition,(
% 2.20/1.21 spl0_1 <=> ((member_a @ y @ carrier) = $true)),
% 2.20/1.21 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 2.20/1.21 thf(f1045,plain,(
% 2.20/1.21 ((member_a @ y @ carrier) = $true)),
% 2.20/1.21 inference(trivial_inequality_removal,[status(thm)],[f1042])).
% 2.20/1.21 thf(f1042,plain,(
% 2.20/1.21 ($true != $true) | ((member_a @ y @ carrier) = $true)),
% 2.20/1.21 inference(superposition,[status(thm)],[f1029,f1017])).
% 2.20/1.21 thf(f1017,plain,(
% 2.20/1.21 ((member449909584od_a_a @ (product_Pair_a_a @ y @ z) @ relation) = $true)),
% 2.20/1.21 inference(cnf_transformation,[status(thm)],[f826])).
% 2.20/1.21 thf(f1041,plain,(
% 2.20/1.21 ~spl0_1 | ~spl0_2),
% 2.20/1.21 inference(avatar_split_clause,[status(thm)],[f1015,f1038,f1034])).
% 2.20/1.21 thf(f1015,plain,(
% 2.20/1.21 ((member_a @ z @ carrier) != $true) | ((member_a @ y @ carrier) != $true)),
% 2.20/1.21 inference(cnf_transformation,[status(thm)],[f1005])).
% 2.20/1.21 thf(f1005,plain,(
% 2.20/1.21 ((member_a @ y @ carrier) != $true) | ((member_a @ z @ carrier) != $true)),
% 2.20/1.21 inference(ennf_transformation,[status(thm)],[f824])).
% 2.20/1.21 thf(f824,plain,(
% 2.20/1.21 ~(((member_a @ y @ carrier) = $true) & ((member_a @ z @ carrier) = $true))),
% 2.20/1.21 inference(fool_elimination,[status(thm)],[f823])).
% 2.20/1.21 thf(f823,plain,(
% 2.20/1.21 ~((member_a @ y @ carrier) & (member_a @ z @ carrier))),
% 2.20/1.21 inference(rectify,[status(thm)],[f357])).
% 2.20/1.21 thf(f357,negated_conjecture,(
% 2.20/1.21 ~((member_a @ y @ carrier) & (member_a @ z @ carrier))),
% 2.20/1.21 inference(negated_conjecture,[status(cth)],[f356])).
% 2.20/1.21 thf(f356,conjecture,(
% 2.20/1.21 (member_a @ y @ carrier) & (member_a @ z @ carrier)),
% 2.20/1.21 file('/export/starexec/sandbox2/tmp/tmp.vxZF8YiGeR/DTF2THF_7300.p',conj_0)).
% 2.20/1.21 % SZS output end Proof for DTF2THF_7300
% 2.20/1.21 % (7517)------------------------------
% 2.20/1.21 % (7517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.21 % (7517)Termination reason: Refutation
% 2.20/1.21
% 2.20/1.21 % (7517)Memory used [KB]: 6524
% 2.20/1.21 % (7517)Time elapsed: 0.020 s
% 2.20/1.21 % (7517)Instructions burned: 60 (million)
% 2.20/1.21 % (7517)------------------------------
% 2.20/1.21 % (7517)------------------------------
% 2.20/1.21 % (7495)Success in time 0.073 s
%------------------------------------------------------------------------------