TPTP Problem File: ITP154^1.p
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%------------------------------------------------------------------------------
% File : ITP154^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_205__6250810_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_205__6250810_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 484 ( 122 unt; 129 typ; 0 def)
% Number of atoms : 1216 ( 241 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 3744 ( 144 ~; 24 |; 133 &;2879 @)
% ( 0 <=>; 564 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 230 ( 230 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 105 usr; 12 con; 0-3 aty)
% Number of variables : 1036 ( 64 ^; 954 !; 18 ?;1036 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:35.897
%------------------------------------------------------------------------------
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prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Set__Oset_Itf__a_J,type,
prefer484977842_set_a: set_set_a > set_Pr2045688071_set_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001tf__a,type,
prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference__axioms_001tf__a,type,
prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc759028718_set_a: produc1691597095_set_a > set_set_a > produc1299534078_set_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
produc1719072398_set_a: produc1691597095_set_a > set_a > produc1472671646_set_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc1599887374_set_a: set_Product_prod_a_a > set_set_a > produc386552094_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_Itf__a_J,type,
produc194376366_set_a: set_Product_prod_a_a > set_a > produc275558334_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc760262016_set_a: set_set_a > produc1691597095_set_a > produc398359056_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc1618062688od_a_a: set_set_a > set_Product_prod_a_a > produc270245360od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc2102896407_set_a: set_set_a > set_set_a > produc407913767_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
produc32105911_set_a: set_set_a > set_a > produc522143687_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc1397177024od_a_a: set_a > set_Product_prod_a_a > produc376942416od_a_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc1560024183_set_a: set_a > set_set_a > produc1083127431_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Relation_OId__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
id_on_1182990020od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(sy_c_Relation_OId__on_001tf__a,type,
id_on_a: set_a > set_Product_prod_a_a ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_Relation_Orefl__on_001tf__a,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_Relation_Ototal__on_001tf__a,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Otrans_001tf__a,type,
trans_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member593450832_set_a: produc1691597095_set_a > set_Pr2045688071_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_carrier,type,
carrier: set_a ).
thf(sy_v_relation,type,
relation: set_Product_prod_a_a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (354)
thf(fact_0_assms_I3_J,axiom,
member_a @ y @ carrier ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member_a @ x @ carrier ).
% assms(2)
thf(fact_2_rational__preference__axioms,axiom,
prefer719835986ence_a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_3_nbt__nest,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% nbt_nest
thf(fact_4_assms_I1_J,axiom,
finite_finite_a @ carrier ).
% assms(1)
thf(fact_5_preference__axioms,axiom,
prefer1292084992ence_a @ carrier @ relation ).
% preference_axioms
thf(fact_6_trans__refl,axiom,
order_preorder_on_a @ carrier @ relation ).
% trans_refl
thf(fact_7_no__better__subset__pref,axiom,
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ).
% no_better_subset_pref
thf(fact_8_no__better__thansubset__rel,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ).
% no_better_thansubset_rel
thf(fact_9_no__better__than__nonepty,axiom,
! [X: a] :
( ( carrier != bot_bot_set_a )
=> ( ( member_a @ X @ carrier )
=> ( ( prefer1676310729than_a @ X @ carrier @ relation )
!= bot_bot_set_a ) ) ) ).
% no_better_than_nonepty
thf(fact_10_total,axiom,
total_on_a @ carrier @ relation ).
% total
thf(fact_11_worse__in__no__better,axiom,
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% worse_in_no_better
thf(fact_12_reflexivity,axiom,
refl_on_a @ carrier @ relation ).
% reflexivity
thf(fact_13_compl,axiom,
! [X2: a] :
( ( member_a @ X2 @ carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% compl
thf(fact_14_completeD,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( X != Y )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% completeD
thf(fact_15_not__outside,axiom,
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( member_a @ X @ carrier ) ) ).
% not_outside
thf(fact_16_strict__is__neg__transitive,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member_a @ X @ carrier )
& ( member_a @ Y @ carrier )
& ( member_a @ Z @ carrier ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ 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_17_weak__is__transitive,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member_a @ X @ carrier )
& ( member_a @ Y @ carrier )
& ( member_a @ Z @ carrier ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation ) ) ) ) ).
% weak_is_transitive
thf(fact_18_strict__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ 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 @ X @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_19_indiff__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ 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 @ X @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_20_strict__not__refl__weak,axiom,
! [X: a,Y: a] :
( ( ( member_a @ X @ carrier )
& ( member_a @ Y @ carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_preference_Onot__outside,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1409946857od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_34_preference_Onot__outside,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer1710114912_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
=> ( member_set_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_35_preference_Onot__outside,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_36_preference_Onot__outside,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( member_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_37_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_38_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_39_preference_Oindiff__trans,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( prefer1710114912_set_a @ Carrier @ Relation )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_40_preference_Oindiff__trans,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: 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 @ X @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_41_preference_Oindiff__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_42_rational__preference_Ocompl,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ! [X2: set_a] :
( ( member_set_a @ X2 @ Carrier )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ Carrier )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X2 @ Xa ) @ Relation )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_43_rational__preference_Ocompl,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ Carrier )
=> ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ Carrier )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_44_rational__preference_Ocompl,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ! [X2: a] :
( ( member_a @ X2 @ Carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ Carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_45_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_46_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_47_rational__preference_OcompleteD,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ X @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
| ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_48_rational__preference_OcompleteD,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ X @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
| ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_49_rational__preference_OcompleteD,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_50_rational__preference_OcompleteD,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_51_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_52_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: 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 @ X @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_53_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ 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 @ X @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_54_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier )
& ( member2057358096od_a_a @ Z @ Carrier ) )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ Z ) @ Relation )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_55_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X @ Carrier )
& ( member_set_a @ Y @ Carrier )
& ( member_set_a @ Z @ Carrier ) )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_56_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier )
& ( member449909584od_a_a @ Z @ Carrier ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_57_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X @ Carrier )
& ( member_a @ Y @ Carrier )
& ( member_a @ Z @ Carrier ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_58_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_59_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X @ Carrier )
& ( member_set_a @ Y @ Carrier ) )
=> ( ( ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
= ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_60_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_61_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X @ Carrier )
& ( member_a @ Y @ Carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_62_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a,Z: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier )
& ( member2057358096od_a_a @ Z @ Carrier ) )
=> ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Z ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ X ) @ 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_63_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member_set_a @ X @ Carrier )
& ( member_set_a @ Y @ Carrier )
& ( member_set_a @ Z @ Carrier ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Z ) @ Relation )
& ~ ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ X ) @ 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_64_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier )
& ( member449909584od_a_a @ Z @ Carrier ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ 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_65_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X @ Carrier )
& ( member_a @ Y @ Carrier )
& ( member_a @ Z @ Carrier ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ 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_66_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( member2057358096od_a_a @ Y @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_67_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
=> ( member_set_a @ Y @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_68_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_69_rational__preference_Oworse__in__no__better,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_70_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo1116205875od_a_a )
=> ( ( member2057358096od_a_a @ X @ Carrier )
=> ( ( prefer926864306od_a_a @ X @ Carrier @ Relation )
!= bot_bo1116205875od_a_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_71_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_set_a )
=> ( ( member_set_a @ X @ Carrier )
=> ( ( prefer1671525929_set_a @ X @ Carrier @ Relation )
!= bot_bot_set_set_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_72_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo2131659635od_a_a )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( prefer1498087410od_a_a @ X @ Carrier @ Relation )
!= bot_bo2131659635od_a_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_73_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_a )
=> ( ( member_a @ X @ Carrier )
=> ( ( prefer1676310729than_a @ X @ Carrier @ Relation )
!= bot_bot_set_a ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_74_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ X @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ( ord_le456379495od_a_a @ ( prefer926864306od_a_a @ Y @ Carrier @ Relation ) @ ( prefer926864306od_a_a @ X @ Carrier @ Relation ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_75_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ X @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_76_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_77_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_78_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
=> ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_79_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_80_rational__preference_Ono__better__subset__pref,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ).
% rational_preference.no_better_subset_pref
thf(fact_81_rational__preference_Onbt__nest,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Y: set_a,X: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) )
| ( ord_le318720350_set_a @ ( prefer1671525929_set_a @ X @ Carrier @ Relation ) @ ( prefer1671525929_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_82_rational__preference_Onbt__nest,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Y: product_prod_a_a,X: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
| ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_83_rational__preference_Onbt__nest,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.nbt_nest
thf(fact_84_at__lst__asgd__not__ge,axiom,
! [X: a,Y: a] :
( ( carrier != bot_bot_set_a )
=> ( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ) ).
% at_lst_asgd_not_ge
thf(fact_85_same__at__least__as__equal,axiom,
! [Z: a,Y: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) )
=> ( ( prefer161218362good_a @ Z @ carrier @ relation )
= ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% same_at_least_as_equal
thf(fact_86_pref__in__at__least__as,axiom,
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% pref_in_at_least_as
thf(fact_87_empty__subsetI,axiom,
! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A ) ).
% empty_subsetI
thf(fact_88_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_89_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_90_subset__empty,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a )
= ( A = bot_bo2131659635od_a_a ) ) ).
% subset_empty
thf(fact_91_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le318720350_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_92_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_93_mem__Collect__eq,axiom,
! [A2: produc1572603623od_a_a,P: produc1572603623od_a_a > $o] :
( ( member2057358096od_a_a @ A2 @ ( collec1635618130od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A: set_Pr1948701895od_a_a] :
( ( collec1635618130od_a_a
@ ^ [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_101_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_102_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_103_Pair__le,axiom,
! [A2: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A2 @ B ) @ ( produc1928581911_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_104_Pair__le,axiom,
! [A2: set_a,B: set_set_a,C: set_a,D: set_set_a] :
( ( ord_le1762684455_set_a @ ( produc1560024183_set_a @ A2 @ B ) @ ( produc1560024183_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_le318720350_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_105_Pair__le,axiom,
! [A2: set_set_a,B: set_a,C: set_set_a,D: set_a] :
( ( ord_le1201700711_set_a @ ( produc32105911_set_a @ A2 @ B ) @ ( produc32105911_set_a @ C @ D ) )
= ( ( ord_le318720350_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_106_Pair__le,axiom,
! [A2: set_a,B: set_Product_prod_a_a,C: set_a,D: set_Product_prod_a_a] :
( ( ord_le49202928od_a_a @ ( produc1397177024od_a_a @ A2 @ B ) @ ( produc1397177024od_a_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_107_Pair__le,axiom,
! [A2: set_Product_prod_a_a,B: set_a,C: set_Product_prod_a_a,D: set_a] :
( ( ord_le2095302494_set_a @ ( produc194376366_set_a @ A2 @ B ) @ ( produc194376366_set_a @ C @ D ) )
= ( ( ord_le1824328871od_a_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_108_Pair__le,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le723713223_set_a @ ( produc2102896407_set_a @ A2 @ B ) @ ( produc2102896407_set_a @ C @ D ) )
= ( ( ord_le318720350_set_a @ A2 @ C )
& ( ord_le318720350_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_109_Pair__le,axiom,
! [A2: set_a,B: produc1691597095_set_a,C: set_a,D: produc1691597095_set_a] :
( ( ord_le560833296_set_a @ ( produc272677600_set_a @ A2 @ B ) @ ( produc272677600_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_le486764743_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_110_Pair__le,axiom,
! [A2: set_Product_prod_a_a,B: set_set_a,C: set_Product_prod_a_a,D: set_set_a] :
( ( ord_le642967742_set_a @ ( produc1599887374_set_a @ A2 @ B ) @ ( produc1599887374_set_a @ C @ D ) )
= ( ( ord_le1824328871od_a_a @ A2 @ C )
& ( ord_le318720350_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_111_Pair__le,axiom,
! [A2: produc1691597095_set_a,B: set_a,C: produc1691597095_set_a,D: set_a] :
( ( ord_le1729087294_set_a @ ( produc1719072398_set_a @ A2 @ B ) @ ( produc1719072398_set_a @ C @ D ) )
= ( ( ord_le486764743_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_112_Pair__le,axiom,
! [A2: set_set_a,B: set_Product_prod_a_a,C: set_set_a,D: set_Product_prod_a_a] :
( ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ A2 @ B ) @ ( produc1618062688od_a_a @ C @ D ) )
= ( ( ord_le318720350_set_a @ A2 @ C )
& ( ord_le1824328871od_a_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_113_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_114_as__good__asIff,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ).
% as_good_asIff
thf(fact_115_finite__has__maximal,axiom,
! [A: set_se1596668135od_a_a] :
( ( finite1145471536od_a_a @ A )
=> ( ( A != bot_bo1424703315od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X4 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ A )
=> ( ( ord_le1824328871od_a_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_116_finite__has__maximal,axiom,
! [A: set_Pr2045688071_set_a] :
( ( finite283787856_set_a @ A )
=> ( ( A != bot_bo671027059_set_a )
=> ? [X4: produc1691597095_set_a] :
( ( member593450832_set_a @ X4 @ A )
& ! [Xa: produc1691597095_set_a] :
( ( member593450832_set_a @ Xa @ A )
=> ( ( ord_le486764743_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_117_finite__has__maximal,axiom,
! [A: set_set_set_a] :
( ( finite1606323175_set_a @ A )
=> ( ( A != bot_bo1264989578_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le318720350_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_118_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_119_finite__has__minimal,axiom,
! [A: set_se1596668135od_a_a] :
( ( finite1145471536od_a_a @ A )
=> ( ( A != bot_bo1424703315od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X4 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ A )
=> ( ( ord_le1824328871od_a_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_120_finite__has__minimal,axiom,
! [A: set_Pr2045688071_set_a] :
( ( finite283787856_set_a @ A )
=> ( ( A != bot_bo671027059_set_a )
=> ? [X4: produc1691597095_set_a] :
( ( member593450832_set_a @ X4 @ A )
& ! [Xa: produc1691597095_set_a] :
( ( member593450832_set_a @ Xa @ A )
=> ( ( ord_le486764743_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_121_finite__has__minimal,axiom,
! [A: set_set_set_a] :
( ( finite1606323175_set_a @ A )
=> ( ( A != bot_bo1264989578_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le318720350_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_122_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_123_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo2131659635od_a_a
= ( collec645855634od_a_a @ P ) )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_124_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_125_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_126_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec645855634od_a_a @ P )
= bot_bo2131659635od_a_a )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_127_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_128_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_129_all__not__in__conv,axiom,
! [A: set_Pr1948701895od_a_a] :
( ( ! [X3: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ X3 @ A ) )
= ( A = bot_bo1116205875od_a_a ) ) ).
% all_not_in_conv
thf(fact_130_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_131_all__not__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member449909584od_a_a @ X3 @ A ) )
= ( A = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_132_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_133_empty__iff,axiom,
! [C: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ C @ bot_bo1116205875od_a_a ) ).
% empty_iff
thf(fact_134_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_135_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_136_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_137_subsetI,axiom,
! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a] :
( ! [X4: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X4 @ A )
=> ( member2057358096od_a_a @ X4 @ B2 ) )
=> ( ord_le456379495od_a_a @ A @ B2 ) ) ).
% subsetI
thf(fact_138_subsetI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B2 ) )
=> ( ord_le318720350_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_139_subsetI,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A )
=> ( member449909584od_a_a @ X4 @ B2 ) )
=> ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% subsetI
thf(fact_140_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_141_subset__antisym,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_142_subset__antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( ord_le318720350_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_143_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_144_at__least__as__goodD,axiom,
! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_145_at__least__as__goodD,axiom,
! [Z: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
( ( member_set_a @ Z @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_146_at__least__as__goodD,axiom,
! [Z: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_147_at__least__as__goodD,axiom,
! [Z: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_148_at__lst__asgd__ge,axiom,
! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B2: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B2 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_149_at__lst__asgd__ge,axiom,
! [X: set_a,Y: set_a,B2: set_set_a,Pr: set_Pr2045688071_set_a] :
( ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ B2 @ Pr ) )
=> ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_150_at__lst__asgd__ge,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_151_at__lst__asgd__ge,axiom,
! [X: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ X @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_152_ex__in__conv,axiom,
! [A: set_Pr1948701895od_a_a] :
( ( ? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A ) )
= ( A != bot_bo1116205875od_a_a ) ) ).
% ex_in_conv
thf(fact_153_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_154_ex__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
= ( A != bot_bo2131659635od_a_a ) ) ).
% ex_in_conv
thf(fact_155_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_156_equals0I,axiom,
! [A: set_Pr1948701895od_a_a] :
( ! [Y3: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ Y3 @ A )
=> ( A = bot_bo1116205875od_a_a ) ) ).
% equals0I
thf(fact_157_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_158_equals0I,axiom,
! [A: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member449909584od_a_a @ Y3 @ A )
=> ( A = bot_bo2131659635od_a_a ) ) ).
% equals0I
thf(fact_159_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_160_equals0D,axiom,
! [A: set_Pr1948701895od_a_a,A2: produc1572603623od_a_a] :
( ( A = bot_bo1116205875od_a_a )
=> ~ ( member2057358096od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_161_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_162_equals0D,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( A = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_163_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_164_emptyE,axiom,
! [A2: produc1572603623od_a_a] :
~ ( member2057358096od_a_a @ A2 @ bot_bo1116205875od_a_a ) ).
% emptyE
thf(fact_165_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_166_emptyE,axiom,
! [A2: product_prod_a_a] :
~ ( member449909584od_a_a @ A2 @ bot_bo2131659635od_a_a ) ).
% emptyE
thf(fact_167_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_168_in__mono,axiom,
! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
( ( ord_le456379495od_a_a @ A @ B2 )
=> ( ( member2057358096od_a_a @ X @ A )
=> ( member2057358096od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_169_in__mono,axiom,
! [A: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_170_in__mono,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B2 )
=> ( ( member449909584od_a_a @ X @ A )
=> ( member449909584od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_171_in__mono,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_172_subsetD,axiom,
! [A: set_Pr1948701895od_a_a,B2: set_Pr1948701895od_a_a,C: produc1572603623od_a_a] :
( ( ord_le456379495od_a_a @ A @ B2 )
=> ( ( member2057358096od_a_a @ C @ A )
=> ( member2057358096od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_173_subsetD,axiom,
! [A: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_174_subsetD,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B2 )
=> ( ( member449909584od_a_a @ C @ A )
=> ( member449909584od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_175_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_176_equalityE,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A = B2 )
=> ~ ( ( ord_le1824328871od_a_a @ A @ B2 )
=> ~ ( ord_le1824328871od_a_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_177_equalityE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ~ ( ( ord_le318720350_set_a @ A @ B2 )
=> ~ ( ord_le318720350_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_178_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_179_subset__eq,axiom,
( ord_le456379495od_a_a
= ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
! [X3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A3 )
=> ( member2057358096od_a_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_180_subset__eq,axiom,
( ord_le318720350_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_181_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A3 )
=> ( member449909584od_a_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_183_equalityD1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A = B2 )
=> ( ord_le1824328871od_a_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_184_equalityD1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ( ord_le318720350_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_185_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_186_equalityD2,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A = B2 )
=> ( ord_le1824328871od_a_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_187_equalityD2,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ( ord_le318720350_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_188_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_189_subset__iff,axiom,
( ord_le456379495od_a_a
= ( ^ [A3: set_Pr1948701895od_a_a,B3: set_Pr1948701895od_a_a] :
! [T: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ T @ A3 )
=> ( member2057358096od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_190_subset__iff,axiom,
( ord_le318720350_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member449909584od_a_a @ T @ A3 )
=> ( member449909584od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__refl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A @ A ) ).
% subset_refl
thf(fact_194_subset__refl,axiom,
! [A: set_set_a] : ( ord_le318720350_set_a @ A @ A ) ).
% subset_refl
thf(fact_195_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_196_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_197_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_198_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_199_subset__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A @ B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ C2 )
=> ( ord_le1824328871od_a_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_200_subset__trans,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( ord_le318720350_set_a @ B2 @ C2 )
=> ( ord_le318720350_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_201_subset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_202_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : Y4 = Z2 )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B3 )
& ( ord_le1824328871od_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_203_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : Y4 = Z2 )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le318720350_set_a @ A3 @ B3 )
& ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_204_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_205_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ ( collec645855634od_a_a @ P ) @ ( collec645855634od_a_a @ Q ) )
= ( ! [X3: product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_206_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_207_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_208_rational__preference_Osame__at__least__as__equal,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,Z: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( ( member593450832_set_a @ ( produc1928581911_set_a @ Z @ Y ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ Z ) @ Relation ) )
=> ( ( prefer1203288218_set_a @ Z @ Carrier @ Relation )
= ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.same_at_least_as_equal
thf(fact_209_rational__preference_Osame__at__least__as__equal,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 )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) )
=> ( ( prefer477097315od_a_a @ Z @ Carrier @ Relation )
= ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.same_at_least_as_equal
thf(fact_210_rational__preference_Osame__at__least__as__equal,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) )
=> ( ( prefer161218362good_a @ Z @ Carrier @ Relation )
= ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.same_at_least_as_equal
thf(fact_211_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_212_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
=> ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_213_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_214_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_215_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_216_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_217_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_218_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_219_rational__preference_Oas__good__asIff,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ X @ ( prefer1867335523od_a_a @ Y @ Carrier @ Relation ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
& ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_220_rational__preference_Oas__good__asIff,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( member_set_a @ X @ ( prefer270414042_set_a @ Y @ Carrier @ Relation ) )
= ( ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation )
& ( member593450832_set_a @ ( produc1928581911_set_a @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_221_rational__preference_Oas__good__asIff,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_222_rational__preference_Oas__good__asIff,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_223_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo1116205875od_a_a )
=> ( ( member2057358096od_a_a @ X @ Carrier )
=> ( ( member2057358096od_a_a @ Y @ Carrier )
=> ( ~ ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ Carrier @ Relation ) )
=> ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_224_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_set_a,Relation: set_Pr2045688071_set_a,X: set_a,Y: set_a] :
( ( prefer484977842_set_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_set_a )
=> ( ( member_set_a @ X @ Carrier )
=> ( ( member_set_a @ Y @ Carrier )
=> ( ~ ( member_set_a @ X @ ( prefer1203288218_set_a @ Y @ Carrier @ Relation ) )
=> ~ ( member593450832_set_a @ ( produc1928581911_set_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_225_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bo2131659635od_a_a )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ~ ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
=> ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_226_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_a )
=> ( ( member_a @ X @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_227_Pair__mono,axiom,
! [X: produc1691597095_set_a,X5: produc1691597095_set_a,Y: set_set_a,Y5: set_set_a] :
( ( ord_le486764743_set_a @ X @ X5 )
=> ( ( ord_le318720350_set_a @ Y @ Y5 )
=> ( ord_le2039441566_set_a @ ( produc759028718_set_a @ X @ Y ) @ ( produc759028718_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_228_Pair__mono,axiom,
! [X: set_set_a,X5: set_set_a,Y: set_a,Y5: set_a] :
( ( ord_le318720350_set_a @ X @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le1201700711_set_a @ ( produc32105911_set_a @ X @ Y ) @ ( produc32105911_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_229_Pair__mono,axiom,
! [X: set_set_a,X5: set_set_a,Y: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le318720350_set_a @ X @ X5 )
=> ( ( ord_le1824328871od_a_a @ Y @ Y5 )
=> ( ord_le526661008od_a_a @ ( produc1618062688od_a_a @ X @ Y ) @ ( produc1618062688od_a_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_230_Pair__mono,axiom,
! [X: set_set_a,X5: set_set_a,Y: produc1691597095_set_a,Y5: produc1691597095_set_a] :
( ( ord_le318720350_set_a @ X @ X5 )
=> ( ( ord_le486764743_set_a @ Y @ Y5 )
=> ( ord_le1138266544_set_a @ ( produc760262016_set_a @ X @ Y ) @ ( produc760262016_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_231_Pair__mono,axiom,
! [X: set_set_a,X5: set_set_a,Y: set_set_a,Y5: set_set_a] :
( ( ord_le318720350_set_a @ X @ X5 )
=> ( ( ord_le318720350_set_a @ Y @ Y5 )
=> ( ord_le723713223_set_a @ ( produc2102896407_set_a @ X @ Y ) @ ( produc2102896407_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_232_Pair__mono,axiom,
! [X: set_a,X5: set_a,Y: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_233_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ A2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_234_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_235_bot__prod__def,axiom,
( bot_bo1519632275_set_a
= ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% bot_prod_def
thf(fact_236_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_237_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_238_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_239_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_240_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_241_old_Oprod_Oinject,axiom,
! [A2: a,B: a,A4: a,B4: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A4 @ B4 ) )
= ( ( A2 = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_242_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_243_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_244_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% preorder_on_empty
thf(fact_245_total__on__def,axiom,
( total_on_a
= ( ^ [A3: set_a,R: set_Product_prod_a_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A3 )
=> ( ( X3 != Y2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_246_total__onI,axiom,
! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A )
=> ( ( member449909584od_a_a @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R2 )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_1490170027od_a_a @ A @ R2 ) ) ).
% total_onI
thf(fact_247_total__onI,axiom,
! [A: set_a,R2: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_on_a @ A @ R2 ) ) ).
% total_onI
thf(fact_248_total__on__empty,axiom,
! [R2: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R2 ) ).
% total_on_empty
thf(fact_249_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_250_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_251_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_252_dual__order_Oantisym,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_253_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_254_dual__order_Otrans,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_255_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_256_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_257_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_class.order.antisym
thf(fact_258_ord__le__eq__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_259_ord__eq__le__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( A2 = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_260_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_261_antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_262_order_Otrans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_263_eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% eq_refl
thf(fact_264_antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_265_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_266_ord__le__eq__subst,axiom,
! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [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 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_267_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_268_order__subst2,axiom,
! [A2: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [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 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_269_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_270_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X4: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_271_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_272_Pair__inject,axiom,
! [A2: a,B: a,A4: a,B4: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A4 @ B4 ) )
=> ~ ( ( A2 = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_273_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A6: a,B6: a] :
( Y
!= ( product_Pair_a_a @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_274_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_275_subrelI,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S2 ) )
=> ( ord_le1824328871od_a_a @ R2 @ S2 ) ) ).
% subrelI
thf(fact_276_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_277_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_278_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_279_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% refl_on_empty
thf(fact_280_refl__onD,axiom,
! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A @ R2 )
=> ( ( member449909584od_a_a @ A2 @ A )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_281_refl__onD,axiom,
! [A: set_a,R2: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ A @ R2 )
=> ( ( member_a @ A2 @ A )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_282_refl__onD1,axiom,
! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( member449909584od_a_a @ X @ A ) ) ) ).
% refl_onD1
thf(fact_283_refl__onD1,axiom,
! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ X @ A ) ) ) ).
% refl_onD1
thf(fact_284_refl__onD2,axiom,
! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( member449909584od_a_a @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_285_refl__onD2,axiom,
! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_286_refl__on__domain,axiom,
! [A: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A2: product_prod_a_a,B: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ R2 )
=> ( ( member449909584od_a_a @ A2 @ A )
& ( member449909584od_a_a @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_287_refl__on__domain,axiom,
! [A: set_a,R2: set_Product_prod_a_a,A2: a,B: a] :
( ( refl_on_a @ A @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
=> ( ( member_a @ A2 @ A )
& ( member_a @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_288_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_289_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_290_finite__transitivity__chain,axiom,
! [A: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
( ( finite179568208od_a_a @ A )
=> ( ! [X4: product_prod_a_a] :
~ ( R3 @ X4 @ X4 )
=> ( ! [X4: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
( ( R3 @ X4 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A )
=> ? [Y6: product_prod_a_a] :
( ( member449909584od_a_a @ Y6 @ A )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A = bot_bo2131659635od_a_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_291_finite__transitivity__chain,axiom,
! [A: set_a,R3: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X4: a] :
~ ( R3 @ X4 @ X4 )
=> ( ! [X4: a,Y3: a,Z3: a] :
( ( R3 @ X4 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Y6: a] :
( ( member_a @ Y6 @ A )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_292_subset__emptyI,axiom,
! [A: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member449909584od_a_a @ X4 @ A )
=> ( ord_le1824328871od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% subset_emptyI
thf(fact_293_subset__emptyI,axiom,
! [A: set_a] :
( ! [X4: a] :
~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_294_ssubst__Pair__rhs,axiom,
! [R2: a,S2: a,R3: set_Product_prod_a_a,S3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_295_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_296_rational__preference__axioms__def,axiom,
prefer580340847ioms_a = total_on_a ).
% rational_preference_axioms_def
thf(fact_297_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_298_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_299_transitivity,axiom,
trans_a @ relation ).
% transitivity
thf(fact_300_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : A3 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_301_preference_Otransitivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( trans_a @ Relation ) ) ).
% preference.transitivity
thf(fact_302_trans__empty,axiom,
trans_a @ bot_bo2131659635od_a_a ).
% trans_empty
thf(fact_303_transD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_304_transE,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_305_transI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X4: a,Y3: a,Z3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R2 ) ) )
=> ( trans_a @ R2 ) ) ).
% transI
thf(fact_306_trans__def,axiom,
( trans_a
= ( ^ [R: set_Product_prod_a_a] :
! [X3: a,Y2: a,Z4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_307_preorder__on__def,axiom,
( order_preorder_on_a
= ( ^ [A3: set_a,R: set_Product_prod_a_a] :
( ( refl_on_a @ A3 @ R )
& ( trans_a @ R ) ) ) ) ).
% preorder_on_def
thf(fact_308_relChain__def,axiom,
( bNF_Ca850758276_set_a
= ( ^ [R: set_Product_prod_a_a,As: a > set_a] :
! [I: a,J: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ I @ J ) @ R )
=> ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_309_Id__on__empty,axiom,
( ( id_on_a @ bot_bot_set_a )
= bot_bo2131659635od_a_a ) ).
% Id_on_empty
thf(fact_310_under__incr,axiom,
! [R2: set_Product_prod_a_a,A2: a,B: a] :
( ( trans_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_a @ ( order_under_a @ R2 @ A2 ) @ ( order_under_a @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_311_Id__onI,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A2 @ A )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ A2 ) @ ( id_on_1182990020od_a_a @ A ) ) ) ).
% Id_onI
thf(fact_312_Id__onI,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( id_on_a @ A ) ) ) ).
% Id_onI
thf(fact_313_trans__Id__on,axiom,
! [A: set_a] : ( trans_a @ ( id_on_a @ A ) ) ).
% trans_Id_on
thf(fact_314_Id__on__iff,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( id_on_1182990020od_a_a @ A ) )
= ( ( X = Y )
& ( member449909584od_a_a @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_315_Id__on__iff,axiom,
! [X: a,Y: a,A: set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( id_on_a @ A ) )
= ( ( X = Y )
& ( member_a @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_316_Id__on__eqI,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ( member449909584od_a_a @ A2 @ A )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A2 @ B ) @ ( id_on_1182990020od_a_a @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_317_Id__on__eqI,axiom,
! [A2: a,B: a,A: set_a] :
( ( A2 = B )
=> ( ( member_a @ A2 @ A )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( id_on_a @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_318_Id__onE,axiom,
! [C: produc1572603623od_a_a,A: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ C @ ( id_on_1182990020od_a_a @ A ) )
=> ~ ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A )
=> ( C
!= ( produc1474507607od_a_a @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_319_Id__onE,axiom,
! [C: product_prod_a_a,A: set_a] :
( ( member449909584od_a_a @ C @ ( id_on_a @ A ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( C
!= ( product_Pair_a_a @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_320_refl__on__Id__on,axiom,
! [A: set_a] : ( refl_on_a @ A @ ( id_on_a @ A ) ) ).
% refl_on_Id_on
thf(fact_321_Greatest__equality,axiom,
! [P: set_a > $o,X: set_a] :
( ( P @ X )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ( order_Greatest_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_322_GreatestI2__order,axiom,
! [P: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P @ X )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [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_323_Inf__fin_Osubset__imp,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ B2 ) @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_324_Sup__fin_Osubset__imp,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le318720350_set_a @ A @ B2 )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ ( lattic501550101_set_a @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_325_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ ( lattic501550101_set_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_326_Sup__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ A2 @ ( lattic501550101_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_327_Inf__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( lattic1600076923_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_328_Inf__fin_OboundedE,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
=> ! [A7: set_a] :
( ( member_set_a @ A7 @ A )
=> ( ord_less_eq_set_a @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_329_Inf__fin_OboundedI,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ( ord_less_eq_set_a @ X @ A6 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_330_Sup__fin_OboundedE,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
=> ! [A7: set_a] :
( ( member_set_a @ A7 @ A )
=> ( ord_less_eq_set_a @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_331_Sup__fin_OboundedI,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ( ord_less_eq_set_a @ A6 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_332_Inf__fin_Obounded__iff,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic1600076923_set_a @ A ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_333_Sup__fin_Obounded__iff,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic501550101_set_a @ A ) @ X )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_334_Un__empty,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_335_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_336_Un__subset__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_337_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_338_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_339_trans__Int,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( trans_a @ R2 )
=> ( ( trans_a @ S2 )
=> ( trans_a @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% trans_Int
thf(fact_340_refl__on__Un,axiom,
! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
( ( refl_on_a @ A @ R2 )
=> ( ( refl_on_a @ B2 @ S2 )
=> ( refl_on_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_su615935707od_a_a @ R2 @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_341_refl__on__Int,axiom,
! [A: set_a,R2: set_Product_prod_a_a,B2: set_a,S2: set_Product_prod_a_a] :
( ( refl_on_a @ A @ R2 )
=> ( ( refl_on_a @ B2 @ S2 )
=> ( refl_on_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_in920712437od_a_a @ R2 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_342_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_343_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_344_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_345_Un__Int__assoc__eq,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_346_Int__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_347_Int__lower1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_348_Int__lower2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_349_Int__absorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_350_Int__absorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_351_Int__greatest,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_352_Int__Collect__mono,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le1824328871od_a_a @ A @ B2 )
=> ( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le1824328871od_a_a @ ( inf_in920712437od_a_a @ A @ ( collec645855634od_a_a @ P ) ) @ ( inf_in920712437od_a_a @ B2 @ ( collec645855634od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_353_Int__Collect__mono,axiom,
! [A: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ) ).
%------------------------------------------------------------------------------