TPTP Problem File: ITP153^1.p
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%------------------------------------------------------------------------------
% File : ITP153^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_180__6250282_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Preferences/prob_180__6250282_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 452 ( 134 unt; 96 typ; 0 def)
% Number of atoms : 1138 ( 279 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 3567 ( 147 ~; 29 |; 123 &;2778 @)
% ( 0 <=>; 490 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 183 ( 183 >; 0 *; 0 +; 0 <<)
% Number of symbols : 86 ( 85 usr; 9 con; 0-3 aty)
% Number of variables : 1027 ( 58 ^; 951 !; 18 ?;1027 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:16.629
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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% Explicit typings (85)
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thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001tf__a,type,
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thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001tf__a,type,
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thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference__axioms_001tf__a,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
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thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001tf__a,type,
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thf(sy_c_Relation_ODomain_001tf__a_001tf__a,type,
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thf(sy_c_Relation_OId__on_001tf__a,type,
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thf(sy_c_Relation_ORange_001tf__a_001tf__a,type,
range_a_a: set_Product_prod_a_a > set_a ).
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thf(sy_c_Relation_Orefl__on_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Relation_Orefl__on_001tf__a,type,
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thf(sy_c_Relation_Ototal__on_001tf__a,type,
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thf(sy_c_Relation_Otrans_001tf__a,type,
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thf(sy_c_Set_OCollect_001tf__a,type,
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thf(sy_c_Set_Oinsert_001tf__a,type,
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thf(sy_c_Set_Ois__empty_001tf__a,type,
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thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_Set_Ois__singleton_001tf__a,type,
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thf(sy_c_Set_Othe__elem_001tf__a,type,
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thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_member_001tf__a,type,
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thf(sy_v_carrier,type,
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thf(sy_v_relation,type,
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thf(sy_v_y,type,
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thf(sy_v_z,type,
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% Relevant facts (355)
thf(fact_0_assms,axiom,
( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ relation )
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% assms
thf(fact_1_rational__preference__axioms,axiom,
prefer719835986ence_a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_2_preference__axioms,axiom,
prefer1292084992ence_a @ carrier @ relation ).
% preference_axioms
thf(fact_3_trans__refl,axiom,
order_preorder_on_a @ carrier @ relation ).
% trans_refl
thf(fact_4_total,axiom,
total_on_a @ carrier @ relation ).
% total
thf(fact_5_reflexivity,axiom,
refl_on_a @ carrier @ relation ).
% reflexivity
thf(fact_6_compl,axiom,
! [X: a] :
( ( member_a @ X @ carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ relation ) ) ) ) ).
% compl
thf(fact_7_completeD,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( X2 != Y )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% completeD
thf(fact_8_not__outside,axiom,
! [X2: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
=> ( member_a @ X2 @ carrier ) ) ).
% not_outside
thf(fact_9_strict__is__neg__transitive,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member_a @ X2 @ carrier )
& ( member_a @ Y @ carrier )
& ( member_a @ Z @ carrier ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% strict_is_neg_transitive
thf(fact_10_weak__is__transitive,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member_a @ X2 @ carrier )
& ( member_a @ Y @ carrier )
& ( member_a @ Z @ carrier ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation ) ) ) ) ).
% weak_is_transitive
thf(fact_11_strict__not__refl__weak,axiom,
! [X2: a,Y: a] :
( ( ( member_a @ X2 @ carrier )
& ( member_a @ Y @ carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_12_indiff__trans,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_13_strict__trans,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_14_pref__in__at__least__as,axiom,
! [X2: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
=> ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% pref_in_at_least_as
thf(fact_15_rational__preference_Oaxioms_I1_J,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_16_rational__preference_Oaxioms_I1_J,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_17_preference__def,axiom,
( prefer1409946857od_a_a
= ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
( ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member2057358096od_a_a @ X3 @ Carrier2 ) )
& ! [X3: produc1572603623od_a_a,Y2: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member2057358096od_a_a @ Y2 @ Carrier2 ) )
& ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_18_preference__def,axiom,
( prefer1710114912_set_a
= ( ^ [Carrier2: set_set_a,Relation2: set_Pr2045688071_set_a] :
( ! [X3: set_a,Y2: set_a] :
( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
=> ( member_set_a @ X3 @ Carrier2 ) )
& ! [X3: set_a,Y2: set_a] :
( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ Relation2 )
=> ( member_set_a @ Y2 @ Carrier2 ) )
& ( order_816746613_set_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_19_preference__def,axiom,
( prefer1338892073od_a_a
= ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member449909584od_a_a @ X3 @ Carrier2 ) )
& ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member449909584od_a_a @ Y2 @ Carrier2 ) )
& ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_20_preference__def,axiom,
( prefer1292084992ence_a
= ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
( ! [X3: a,Y2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member_a @ X3 @ Carrier2 ) )
& ! [X3: a,Y2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
=> ( member_a @ Y2 @ Carrier2 ) )
& ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_21_preference_Ointro,axiom,
! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ X4 @ Carrier ) )
=> ( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ Y3 @ Carrier ) )
=> ( ( order_491428350od_a_a @ Carrier @ Relation )
=> ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_22_preference_Ointro,axiom,
! [Relation: set_Pr2045688071_set_a,Carrier: set_set_a] :
( ! [X4: set_a,Y3: set_a] :
( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
=> ( member_set_a @ X4 @ Carrier ) )
=> ( ! [X4: set_a,Y3: set_a] :
( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ Relation )
=> ( member_set_a @ Y3 @ Carrier ) )
=> ( ( order_816746613_set_a @ Carrier @ Relation )
=> ( prefer1710114912_set_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_23_preference_Ointro,axiom,
! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ X4 @ Carrier ) )
=> ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ Y3 @ Carrier ) )
=> ( ( order_359341630od_a_a @ Carrier @ Relation )
=> ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_24_preference_Ointro,axiom,
! [Relation: set_Product_prod_a_a,Carrier: set_a] :
( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
=> ( member_a @ X4 @ Carrier ) )
=> ( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
=> ( member_a @ Y3 @ Carrier ) )
=> ( ( order_preorder_on_a @ Carrier @ Relation )
=> ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_25_preference_Otrans__refl,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_26_preference_Otrans__refl,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_27_preference_Onot__outside,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1409946857od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X2 @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_28_preference_Onot__outside,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer1710114912_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
=> ( member_set_a @ X2 @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_29_preference_Onot__outside,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
=> ( member449909584od_a_a @ X2 @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_30_preference_Onot__outside,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
=> ( member_a @ X2 @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_31_preference_Oreflexivity,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_32_preference_Oreflexivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( refl_on_a @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_33_preference_Oindiff__trans,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
( ( prefer1710114912_set_a @ Carrier @ Relation )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_34_preference_Oindiff__trans,axiom,
! [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] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_35_preference_Oindiff__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_36_rational__preference_Ocompl,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ! [X: set_a] :
( ( member_set_a @ X @ Carrier )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ Carrier )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Xa ) @ Relation )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_37_rational__preference_Ocompl,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ! [X: product_prod_a_a] :
( ( member449909584od_a_a @ X @ Carrier )
=> ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ Carrier )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Xa ) @ Relation )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_38_rational__preference_Ocompl,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ! [X: a] :
( ( member_a @ X @ Carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ Carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ Relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_39_rational__preference_Ototal,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_40_rational__preference_Ototal,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( total_on_a @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_41_rational__preference_OcompleteD,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ X2 @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ( X2 != Y )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
| ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_42_rational__preference_OcompleteD,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ X2 @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ( X2 != Y )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_43_rational__preference_OcompleteD,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X2 @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( X2 != Y )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_44_rational__preference_OcompleteD,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X2 @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ( X2 != Y )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_45_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation ) )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_46_rational__preference_Ostrict__trans,axiom,
! [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] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_47_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_48_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X2 @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier )
& ( member2057358096od_a_a @ Z @ Carrier ) )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_49_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X2 @ Carrier )
& ( member_set_a @ Y @ Carrier )
& ( member_set_a @ Z @ Carrier ) )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_50_rational__preference_Oweak__is__transitive,axiom,
! [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] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X2 @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier )
& ( member449909584od_a_a @ Z @ Carrier ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_51_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X2 @ Carrier )
& ( member_a @ Y @ Carrier )
& ( member_a @ Z @ Carrier ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_52_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X2 @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_53_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X2 @ Carrier )
& ( member_set_a @ Y @ Carrier ) )
=> ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
= ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_54_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X2 @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_55_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X2 @ Carrier )
& ( member_a @ Y @ Carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_56_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X2 @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier )
& ( member2057358096od_a_a @ Z @ Carrier ) )
=> ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Z ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X2 ) @ Relation ) )
| ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_57_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X2 @ Carrier )
& ( member_set_a @ Y @ Carrier )
& ( member_set_a @ Z @ Carrier ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Z ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X2 ) @ Relation ) )
| ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_58_rational__preference_Ostrict__is__neg__transitive,axiom,
! [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] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X2 @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier )
& ( member449909584od_a_a @ Z @ Carrier ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X2 ) @ Relation ) )
| ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_59_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X2 @ Carrier )
& ( member_a @ Y @ Carrier )
& ( member_a @ Z @ Carrier ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ Relation ) )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_60_worse__in__no__better,axiom,
! [X2: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
=> ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% worse_in_no_better
thf(fact_61_as__good__as__sameIff,axiom,
! [Z: a,Y: a] :
( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% as_good_as_sameIff
thf(fact_62_prod_Oinject,axiom,
! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y22: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ X1 @ X22 )
= ( produc1474507607od_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_63_prod_Oinject,axiom,
! [X1: set_a,X22: set_a,Y1: set_a,Y22: set_a] :
( ( ( produc1928581911_set_a @ X1 @ X22 )
= ( produc1928581911_set_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_64_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_65_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B )
= ( produc1474507607od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_66_old_Oprod_Oinject,axiom,
! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
( ( ( produc1928581911_set_a @ A @ B )
= ( produc1928581911_set_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_68_rational__preference_Oas__good__as__sameIff,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,Z: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ Z @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Relation )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_69_rational__preference_Oas__good__as__sameIff,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ Z @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
= ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_70_rational__preference_Oas__good__as__sameIff,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_71_rational__preference_Oas__good__as__sameIff,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_72_total__onI,axiom,
! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a] :
( ! [X4: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A3 )
=> ( ( member2057358096od_a_a @ Y3 @ A3 )
=> ( ( X4 != Y3 )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y3 ) @ R )
| ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_1340260971od_a_a @ A3 @ R ) ) ).
% total_onI
thf(fact_73_total__onI,axiom,
! [A3: set_set_a,R: set_Pr2045688071_set_a] :
( ! [X4: set_a,Y3: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( ( X4 != Y3 )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X4 @ Y3 ) @ R )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_on_set_a @ A3 @ R ) ) ).
% total_onI
thf(fact_74_total__onI,axiom,
! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A3 )
=> ( ( member449909584od_a_a @ Y3 @ A3 )
=> ( ( X4 != Y3 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_1490170027od_a_a @ A3 @ R ) ) ).
% total_onI
thf(fact_75_total__onI,axiom,
! [A3: set_a,R: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( ( X4 != Y3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_on_a @ A3 @ R ) ) ).
% total_onI
thf(fact_76_total__on__def,axiom,
( total_on_set_a
= ( ^ [A4: set_set_a,R2: set_Pr2045688071_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X3 @ Y2 ) @ R2 )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_77_total__on__def,axiom,
( total_1490170027od_a_a
= ( ^ [A4: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ! [Y2: product_prod_a_a] :
( ( member449909584od_a_a @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ R2 )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_78_total__on__def,axiom,
( total_on_a
= ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_79_refl__onD,axiom,
! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A3 @ R )
=> ( ( member2057358096od_a_a @ A @ A3 )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_80_refl__onD,axiom,
! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a] :
( ( refl_on_set_a @ A3 @ R )
=> ( ( member_set_a @ A @ A3 )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_81_refl__onD,axiom,
! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A3 @ R )
=> ( ( member449909584od_a_a @ A @ A3 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_82_refl__onD,axiom,
! [A3: set_a,R: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A3 @ R )
=> ( ( member_a @ A @ A3 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_83_refl__onD1,axiom,
! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A3 @ R )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
=> ( member2057358096od_a_a @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_84_refl__onD1,axiom,
! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( refl_on_set_a @ A3 @ R )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
=> ( member_set_a @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_85_refl__onD1,axiom,
! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A3 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
=> ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_86_refl__onD1,axiom,
! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
( ( refl_on_a @ A3 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( member_a @ X2 @ A3 ) ) ) ).
% refl_onD1
thf(fact_87_refl__onD2,axiom,
! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A3 @ R )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ R )
=> ( member2057358096od_a_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_88_refl__onD2,axiom,
! [A3: set_set_a,R: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( refl_on_set_a @ A3 @ R )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ R )
=> ( member_set_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_89_refl__onD2,axiom,
! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A3 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ R )
=> ( member449909584od_a_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_90_refl__onD2,axiom,
! [A3: set_a,R: set_Product_prod_a_a,X2: a,Y: a] :
( ( refl_on_a @ A3 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( member_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_91_refl__on__domain,axiom,
! [A3: set_Pr1948701895od_a_a,R: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A3 @ R )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R )
=> ( ( member2057358096od_a_a @ A @ A3 )
& ( member2057358096od_a_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_92_refl__on__domain,axiom,
! [A3: set_set_a,R: set_Pr2045688071_set_a,A: set_a,B: set_a] :
( ( refl_on_set_a @ A3 @ R )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ A @ B ) @ R )
=> ( ( member_set_a @ A @ A3 )
& ( member_set_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_93_refl__on__domain,axiom,
! [A3: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A3 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
=> ( ( member449909584od_a_a @ A @ A3 )
& ( member449909584od_a_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_94_refl__on__domain,axiom,
! [A3: set_a,R: set_Product_prod_a_a,A: a,B: a] :
( ( refl_on_a @ A3 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( ( member_a @ A @ A3 )
& ( member_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_95_at__least__as__goodD,axiom,
! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_96_at__least__as__goodD,axiom,
! [Z: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_97_at__least__as__goodD,axiom,
! [Z: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_98_at__least__as__goodD,axiom,
! [Z: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_99_at__lst__asgd__ge,axiom,
! [X2: produc1572603623od_a_a,Y: produc1572603623od_a_a,B3: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ B3 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_100_at__lst__asgd__ge,axiom,
! [X2: set_a,Y: set_a,B3: set_set_a,Pr: set_Pr2045688071_set_a] :
( ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ B3 @ Pr ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_101_at__lst__asgd__ge,axiom,
! [X2: product_prod_a_a,Y: product_prod_a_a,B3: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ B3 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_102_at__lst__asgd__ge,axiom,
! [X2: a,Y: a,B3: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ X2 @ ( prefer161218362good_a @ Y @ B3 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_103_old_Oprod_Oinducts,axiom,
! [P: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_104_old_Oprod_Oinducts,axiom,
! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_105_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_106_old_Oprod_Oexhaust,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A5: product_prod_a_a,B4: product_prod_a_a] :
( Y
!= ( produc1474507607od_a_a @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_107_old_Oprod_Oexhaust,axiom,
! [Y: produc1691597095_set_a] :
~ ! [A5: set_a,B4: set_a] :
( Y
!= ( produc1928581911_set_a @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_108_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A5: a,B4: a] :
( Y
!= ( product_Pair_a_a @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_109_mem__Collect__eq,axiom,
! [A: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A3: set_Pr1948701895od_a_a] :
( ( collec1635618130od_a_a
@ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_115_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec645855634od_a_a @ P )
= ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_116_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_117_prod__induct3,axiom,
! [P: produc1572603623od_a_a > $o,X2: produc1572603623od_a_a] :
( ! [A5: product_prod_a_a,B4: a,C: a] : ( P @ ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_118_prod__cases3,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A5: product_prod_a_a,B4: a,C: a] :
( Y
!= ( produc1474507607od_a_a @ A5 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_119_Pair__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B )
= ( produc1474507607od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_120_Pair__inject,axiom,
! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
( ( ( produc1928581911_set_a @ A @ B )
= ( produc1928581911_set_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_121_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_122_prod__cases,axiom,
! [P: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a] :
( ! [A5: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc1474507607od_a_a @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_123_prod__cases,axiom,
! [P: produc1691597095_set_a > $o,P2: produc1691597095_set_a] :
( ! [A5: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_124_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A5: a,B4: a] : ( P @ ( product_Pair_a_a @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_125_surj__pair,axiom,
! [P2: produc1572603623od_a_a] :
? [X4: product_prod_a_a,Y3: product_prod_a_a] :
( P2
= ( produc1474507607od_a_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_126_surj__pair,axiom,
! [P2: produc1691597095_set_a] :
? [X4: set_a,Y3: set_a] :
( P2
= ( produc1928581911_set_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_127_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X4: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_128_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
=> ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_129_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
=> ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_130_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
=> ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_131_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
=> ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_132_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_133_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
=> ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_134_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
=> ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_135_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
=> ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_136_at__lst__asgd__not__ge,axiom,
! [X2: a,Y: a] :
( ( carrier != bot_bot_set_a )
=> ( ( member_a @ X2 @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% at_lst_asgd_not_ge
thf(fact_137_no__better__subset__pref,axiom,
! [X2: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% no_better_subset_pref
thf(fact_138_no__better__thansubset__rel,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ relation ) ) ) ) ).
% no_better_thansubset_rel
thf(fact_139_no__better__than__nonepty,axiom,
! [X2: a] :
( ( carrier != bot_bot_set_a )
=> ( ( member_a @ X2 @ carrier )
=> ( ( prefer1676310729than_a @ X2 @ carrier @ relation )
!= bot_bot_set_a ) ) ) ).
% no_better_than_nonepty
thf(fact_140_nbt__nest,axiom,
! [Y: a,X2: a] :
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% nbt_nest
thf(fact_141_rational__preference_Ointro,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( prefer1064779160od_a_a @ Carrier @ Relation )
=> ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_142_rational__preference_Ointro,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( prefer580340847ioms_a @ Carrier @ Relation )
=> ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_143_rational__preference__def,axiom,
( prefer697821563od_a_a
= ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
& ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_144_rational__preference__def,axiom,
( prefer719835986ence_a
= ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
& ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_145_refl__on__empty,axiom,
refl_o1298442278od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% refl_on_empty
thf(fact_146_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% refl_on_empty
thf(fact_147_preorder__on__empty,axiom,
order_359341630od_a_a @ bot_bo2131659635od_a_a @ bot_bo1116205875od_a_a ).
% preorder_on_empty
thf(fact_148_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% preorder_on_empty
thf(fact_149_total__on__empty,axiom,
! [R: set_Pr1948701895od_a_a] : ( total_1490170027od_a_a @ bot_bo2131659635od_a_a @ R ) ).
% total_on_empty
thf(fact_150_total__on__empty,axiom,
! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% total_on_empty
thf(fact_151_rational__preference_Onbt__nest,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X2: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
| ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_152_rational__preference_Onbt__nest,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X2: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_153_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo1116205875od_a_a )
=> ( ( member2057358096od_a_a @ X2 @ Carrier )
=> ( ( prefer926864306od_a_a @ X2 @ Carrier @ Relation )
!= bot_bo1116205875od_a_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_154_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo2131659635od_a_a )
=> ( ( member449909584od_a_a @ X2 @ Carrier )
=> ( ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation )
!= bot_bo2131659635od_a_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_155_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_a )
=> ( ( member_a @ X2 @ Carrier )
=> ( ( prefer1676310729than_a @ X2 @ Carrier @ Relation )
!= bot_bot_set_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_156_rational__preference__axioms__def,axiom,
prefer1064779160od_a_a = total_1490170027od_a_a ).
% rational_preference_axioms_def
thf(fact_157_rational__preference__axioms__def,axiom,
prefer580340847ioms_a = total_on_a ).
% rational_preference_axioms_def
thf(fact_158_rational__preference__axioms_Ointro,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( total_1490170027od_a_a @ Carrier @ Relation )
=> ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_159_rational__preference__axioms_Ointro,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( total_on_a @ Carrier @ Relation )
=> ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_160_rational__preference_Oaxioms_I2_J,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_161_rational__preference_Oaxioms_I2_J,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_162_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation )
=> ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_163_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation )
=> ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_164_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_165_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ X2 @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X2 @ Carrier @ Relation ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_166_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ X2 @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X2 @ Carrier @ Relation ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_167_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X2 @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_168_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X2 @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_169_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X2: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo1116205875od_a_a )
=> ( ( member2057358096od_a_a @ X2 @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ~ ( member2057358096od_a_a @ X2 @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
=> ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X2: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_set_a )
=> ( ( member_set_a @ X2 @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ~ ( member_set_a @ X2 @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
=> ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_171_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X2: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo2131659635od_a_a )
=> ( ( member449909584od_a_a @ X2 @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ~ ( member449909584od_a_a @ X2 @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_172_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X2: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_a )
=> ( ( member_a @ X2 @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ~ ( member_a @ X2 @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_173_subset__empty,axiom,
! [A3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
= ( A3 = bot_bo2131659635od_a_a ) ) ).
% subset_empty
thf(fact_174_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_175_empty__subsetI,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% empty_subsetI
thf(fact_176_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_177_Pair__le,axiom,
! [A: set_a,B: set_a,C2: set_a,D: set_a] :
( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C2 @ D ) )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_178_subsetI,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A3 )
=> ( member449909584od_a_a @ X4 @ B3 ) )
=> ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_179_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_180_subset__antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_181_empty__iff,axiom,
! [C2: product_prod_a_a] :
~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_182_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_183_all__not__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member449909584od_a_a @ X3 @ A3 ) )
= ( A3 = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_184_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_185_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_186_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_187_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S ) )
=> ( ord_le1824328871od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_188_ex__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
= ( A3 != bot_bo2131659635od_a_a ) ) ).
% ex_in_conv
thf(fact_189_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_190_equals0I,axiom,
! [A3: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member449909584od_a_a @ Y3 @ A3 )
=> ( A3 = bot_bo2131659635od_a_a ) ) ).
% equals0I
thf(fact_191_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_192_equals0D,axiom,
! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A3 = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% equals0D
thf(fact_193_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_194_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% emptyE
thf(fact_195_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_196_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_197_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_198_subset__trans,axiom,
! [A3: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_199_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_200_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_201_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member449909584od_a_a @ T @ A4 )
=> ( member449909584od_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_202_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_203_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_204_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_205_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A4 )
=> ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_206_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_207_equalityE,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_208_subsetD,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B3 )
=> ( ( member449909584od_a_a @ C2 @ A3 )
=> ( member449909584od_a_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_209_subsetD,axiom,
! [A3: set_a,B3: set_a,C2: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_210_in__mono,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B3 )
=> ( ( member449909584od_a_a @ X2 @ A3 )
=> ( member449909584od_a_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_211_in__mono,axiom,
! [A3: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_212_Pair__mono,axiom,
! [X2: set_a,X5: set_a,Y: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_213_bot__prod__def,axiom,
( bot_bo1519632275_set_a
= ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% bot_prod_def
thf(fact_214_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_215_subset__emptyI,axiom,
! [A3: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member449909584od_a_a @ X4 @ A3 )
=> ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a ) ) ).
% subset_emptyI
thf(fact_216_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X4: a] :
~ ( member_a @ X4 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_217_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_218_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_219_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_220_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_221_bot__empty__eq,axiom,
( bot_bo1293121322_a_a_o
= ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_222_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_223_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_224_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_225_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_226_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_227_eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_228_antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_229_eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% eq_refl
thf(fact_230_order_Otrans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_231_antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_232_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_233_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_234_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_235_order__class_Oorder_Oantisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_236_order__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_237_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_238_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_239_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_240_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_241_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R3: set_Product_prod_a_a,S2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_242_transitivity,axiom,
trans_a @ relation ).
% transitivity
thf(fact_243_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : A4 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_244_transD,axiom,
! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( trans_a @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_245_transE,axiom,
! [R: set_Product_prod_a_a,X2: a,Y: a,Z: a] :
( ( trans_a @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_246_transI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X4: a,Y3: a,Z3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
=> ( trans_a @ R ) ) ).
% transI
thf(fact_247_trans__def,axiom,
( trans_a
= ( ^ [R2: set_Product_prod_a_a] :
! [X3: a,Y2: a,Z4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% trans_def
thf(fact_248_trans__empty,axiom,
trans_a @ bot_bo2131659635od_a_a ).
% trans_empty
thf(fact_249_preference_Otransitivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( trans_a @ Relation ) ) ).
% preference.transitivity
thf(fact_250_preorder__on__def,axiom,
( order_preorder_on_a
= ( ^ [A4: set_a,R2: set_Product_prod_a_a] :
( ( refl_on_a @ A4 @ R2 )
& ( trans_a @ R2 ) ) ) ) ).
% preorder_on_def
thf(fact_251_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_252_relChain__def,axiom,
( bNF_Ca850758276_set_a
= ( ^ [R2: set_Product_prod_a_a,As: a > set_a] :
! [I: a,J: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 )
=> ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_253_Id__on__empty,axiom,
( ( id_on_a @ bot_bot_set_a )
= bot_bo2131659635od_a_a ) ).
% Id_on_empty
thf(fact_254_Id__onI,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A3 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ).
% Id_onI
thf(fact_255_Id__onI,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A3 ) ) ) ).
% Id_onI
thf(fact_256_Id__on__iff,axiom,
! [X2: product_prod_a_a,Y: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Y ) @ ( id_on_1182990020od_a_a @ A3 ) )
= ( ( X2 = Y )
& ( member449909584od_a_a @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_257_Id__on__iff,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( id_on_a @ A3 ) )
= ( ( X2 = Y )
& ( member_a @ X2 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_258_Id__on__eqI,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( A = B )
=> ( ( member449909584od_a_a @ A @ A3 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ ( id_on_1182990020od_a_a @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_259_Id__on__eqI,axiom,
! [A: a,B: a,A3: set_a] :
( ( A = B )
=> ( ( member_a @ A @ A3 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_260_Id__onE,axiom,
! [C2: produc1572603623od_a_a,A3: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ C2 @ ( id_on_1182990020od_a_a @ A3 ) )
=> ~ ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A3 )
=> ( C2
!= ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_261_Id__onE,axiom,
! [C2: product_prod_a_a,A3: set_a] :
( ( member449909584od_a_a @ C2 @ ( id_on_a @ A3 ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( C2
!= ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_262_trans__Id__on,axiom,
! [A3: set_a] : ( trans_a @ ( id_on_a @ A3 ) ) ).
% trans_Id_on
thf(fact_263_refl__on__Id__on,axiom,
! [A3: set_a] : ( refl_on_a @ A3 @ ( id_on_a @ A3 ) ) ).
% refl_on_Id_on
thf(fact_264_under__incr,axiom,
! [R: set_Product_prod_a_a,A: a,B: a] :
( ( trans_a @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( ord_less_eq_set_a @ ( order_under_a @ R @ A ) @ ( order_under_a @ R @ B ) ) ) ) ).
% under_incr
thf(fact_265_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ! [X4: set_a] :
( ( P @ X4 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_266_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_267_trans__singleton,axiom,
! [A: a] : ( trans_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ A ) @ bot_bo2131659635od_a_a ) ) ).
% trans_singleton
thf(fact_268_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_269_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
= ( ( A = B )
| ( member449909584od_a_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_270_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_271_insertCI,axiom,
! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member449909584od_a_a @ A @ B3 )
=> ( A = B ) )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_272_singletonI,axiom,
! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% singletonI
thf(fact_273_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_274_insert__subset,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ A3 ) @ B3 )
= ( ( member449909584od_a_a @ X2 @ B3 )
& ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_275_insert__subset,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B3 )
= ( ( member_a @ X2 @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_276_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_277_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_278_prod_Ocollapse,axiom,
! [Prod: product_prod_a_a] :
( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_279_insert__subsetI,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,X6: set_Product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A3 )
=> ( ( ord_le1824328871od_a_a @ X6 @ A3 )
=> ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X2 @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_280_insert__subsetI,axiom,
! [X2: a,A3: set_a,X6: set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ X6 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_281_less__eq__prod__def,axiom,
( ord_le486764743_set_a
= ( ^ [X3: produc1691597095_set_a,Y2: produc1691597095_set_a] :
( ( ord_less_eq_set_a @ ( produc1926412547_set_a @ X3 ) @ ( produc1926412547_set_a @ Y2 ) )
& ( ord_less_eq_set_a @ ( produc918972485_set_a @ X3 ) @ ( produc918972485_set_a @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_282_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_283_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_284_doubleton__eq__iff,axiom,
! [A: a,B: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_285_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_286_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_287_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member449909584od_a_a @ B @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_288_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_289_subset__insertI2,axiom,
! [A3: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_290_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_291_subset__insert,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ X2 @ A3 )
=> ( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X2 @ B3 ) )
= ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_292_subset__insert,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B3 ) )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_293_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_294_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B7: set_a] :
( ( A3
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_295_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A3 )
=> ? [B7: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ A @ B7 ) )
& ~ ( member449909584od_a_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_296_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A3
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_297_insert__eq__iff,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B3: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ A3 )
=> ( ~ ( member449909584od_a_a @ B @ B3 )
=> ( ( ( insert1116662519od_a_a @ A @ A3 )
= ( insert1116662519od_a_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C4: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ B @ C4 ) )
& ~ ( member449909584od_a_a @ B @ C4 )
& ( B3
= ( insert1116662519od_a_a @ A @ C4 ) )
& ~ ( member449909584od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_298_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_299_insert__absorb,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A3 )
=> ( ( insert1116662519od_a_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_300_insert__ident,axiom,
! [X2: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B3 )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_301_insert__ident,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ X2 @ A3 )
=> ( ~ ( member449909584od_a_a @ X2 @ B3 )
=> ( ( ( insert1116662519od_a_a @ X2 @ A3 )
= ( insert1116662519od_a_a @ X2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_302_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B7: set_a] :
( ( A3
= ( insert_a @ X2 @ B7 ) )
=> ( member_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_303_Set_Oset__insert,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A3 )
=> ~ ! [B7: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ X2 @ B7 ) )
=> ( member449909584od_a_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_304_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_305_insertI2,axiom,
! [A: product_prod_a_a,B3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member449909584od_a_a @ A @ B3 )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_306_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_307_insertI1,axiom,
! [A: product_prod_a_a,B3: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B3 ) ) ).
% insertI1
thf(fact_308_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_309_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member449909584od_a_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_310_surjective__pairing,axiom,
! [T2: product_prod_a_a] :
( T2
= ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).
% surjective_pairing
thf(fact_311_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_a] :
( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_312_fst__conv,axiom,
! [X1: a,X22: a] :
( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_313_fst__eqD,axiom,
! [X2: a,Y: a,A: a] :
( ( ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_314_snd__conv,axiom,
! [X1: a,X22: a] :
( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_315_snd__eqD,axiom,
! [X2: a,Y: a,A: a] :
( ( ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_316_refl__on__singleton,axiom,
! [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 ) ) ).
% refl_on_singleton
thf(fact_317_total__on__singleton,axiom,
! [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 ) ) ).
% total_on_singleton
thf(fact_318_subset__singletonD,axiom,
! [A3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_319_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_320_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > a > $o,X2: a,Y: a,A: product_prod_a_a] :
( ( P @ X2 @ Y )
=> ( ( A
= ( product_Pair_a_a @ X2 @ Y ) )
=> ( P @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_321_conjI__realizer,axiom,
! [P: a > $o,P2: a,Q: a > $o,Q2: a] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_322_exI__realizer,axiom,
! [P: a > a > $o,Y: a,X2: a] :
( ( P @ Y @ X2 )
=> ( P @ ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_323_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_324_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_325_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_326_is__singletonI_H,axiom,
! [A3: set_Product_prod_a_a] :
( ( A3 != bot_bo2131659635od_a_a )
=> ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A3 )
=> ( ( member449909584od_a_a @ Y3 @ A3 )
=> ( X4 = Y3 ) ) )
=> ( is_sin877026331od_a_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_327_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_328_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X4: a] :
( A3
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_329_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_330_linear__order__on__singleton,axiom,
! [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 ) ) ).
% linear_order_on_singleton
thf(fact_331_sndI,axiom,
! [X2: product_prod_a_a,Y: a,Z: a] :
( ( X2
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_snd_a_a @ X2 )
= Z ) ) ).
% sndI
thf(fact_332_lnear__order__on__empty,axiom,
order_653178931r_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% lnear_order_on_empty
thf(fact_333_fstI,axiom,
! [X2: product_prod_a_a,Y: a,Z: a] :
( ( X2
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_fst_a_a @ X2 )
= Y ) ) ).
% fstI
thf(fact_334_eq__snd__iff,axiom,
! [B: a,P2: product_prod_a_a] :
( ( B
= ( product_snd_a_a @ P2 ) )
= ( ? [A6: a] :
( P2
= ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_335_eq__fst__iff,axiom,
! [A: a,P2: product_prod_a_a] :
( ( A
= ( product_fst_a_a @ P2 ) )
= ( ? [B6: a] :
( P2
= ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_336_prod_Oswap__def,axiom,
( product_swap_a_a
= ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_337_Range__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( range_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( insert_a @ B @ ( range_a_a @ R ) ) ) ).
% Range_insert
thf(fact_338_swap__simp,axiom,
! [X2: a,Y: a] :
( ( product_swap_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
= ( product_Pair_a_a @ Y @ X2 ) ) ).
% swap_simp
thf(fact_339_RangeE,axiom,
! [B: a,R: set_Product_prod_a_a] :
( ( member_a @ B @ ( range_a_a @ R ) )
=> ~ ! [A5: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B ) @ R ) ) ).
% RangeE
thf(fact_340_Range__iff,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
= ( ? [Y2: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_341_Range_Ocases,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
=> ~ ! [A5: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ A ) @ R ) ) ).
% Range.cases
thf(fact_342_Range_Osimps,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
= ( ? [A6: a,B6: a] :
( ( A = B6 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_343_Range_Ointros,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member_a @ B @ ( range_a_a @ R ) ) ) ).
% Range.intros
thf(fact_344_Range_Oinducts,axiom,
! [X2: a,R: set_Product_prod_a_a,P: a > $o] :
( ( member_a @ X2 @ ( range_a_a @ R ) )
=> ( ! [A5: a,B4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ B4 ) @ R )
=> ( P @ B4 ) )
=> ( P @ X2 ) ) ) ).
% Range.inducts
thf(fact_345_Domain__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( domain_a_a @ ( insert1116662519od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( insert_a @ A @ ( domain_a_a @ R ) ) ) ).
% Domain_insert
thf(fact_346_subset__Compl__singleton,axiom,
! [A3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ ( uminus1440301054od_a_a @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) )
= ( ~ ( member449909584od_a_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_347_subset__Compl__singleton,axiom,
! [A3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_348_Compl__iff,axiom,
! [C2: a,A3: set_a] :
( ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) )
= ( ~ ( member_a @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_349_Compl__iff,axiom,
! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) )
= ( ~ ( member449909584od_a_a @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_350_ComplI,axiom,
! [C2: a,A3: set_a] :
( ~ ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_351_ComplI,axiom,
! [C2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ C2 @ A3 )
=> ( member449909584od_a_a @ C2 @ ( uminus1440301054od_a_a @ A3 ) ) ) ).
% ComplI
thf(fact_352_Compl__anti__mono,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_353_Compl__subset__Compl__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B3 ) )
= ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_354_subset__Compl__self__eq,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
% Conjectures (1)
thf(conj_0,conjecture,
( ( member_a @ z @ carrier )
& ( member_a @ y @ carrier ) ) ).
%------------------------------------------------------------------------------