TPTP Problem File: ITP155^1.p
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%------------------------------------------------------------------------------
% File : ITP155^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_260__6251520_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_260__6251520_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 443 ( 98 unt; 88 typ; 0 def)
% Number of atoms : 1290 ( 214 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 4010 ( 124 ~; 26 |; 135 &;3019 @)
% ( 0 <=>; 706 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 184 ( 184 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 74 usr; 11 con; 0-3 aty)
% Number of variables : 1089 ( 61 ^; 986 !; 42 ?;1089 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:45:04.162
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_Pr1948701895od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc1572603623od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1691597095_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
produc1404140509_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
produc1463940355_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (74)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite1481642319od_a_a: set_Product_prod_a_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite179568208od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
order_1150839801on_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001tf__a,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
bot_bo1519632275_set_a: produc1691597095_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
ord_le841160035_set_a: produc1463940355_set_a > produc1463940355_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
ord_le781360189_a_nat: produc1404140509_a_nat > produc1404140509_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Nat__Onat,type,
prefer1936906324as_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001tf__a,type,
prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Nat__Onat,type,
prefer563798164od_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001tf__a,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Nat__Onat,type,
prefer1672638789an_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001tf__a,type,
prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Nat__Onat,type,
prefer1293349710ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001tf__a,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Nat__Onat,type,
prefer1147844220ce_nat: set_nat > set_Pr1986765409at_nat > $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_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__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
produc1450889781_set_a: nat > set_a > produc1463940355_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_Itf__a_J_001t__Nat__Onat,type,
produc515611863_a_nat: set_a > nat > produc1404140509_a_nat ).
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_Orefl__on_001t__Nat__Onat,type,
refl_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
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__Nat__Onat,type,
total_on_nat: set_nat > set_Pr1986765409at_nat > $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__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
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_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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $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_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_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_I2_J,axiom,
member_a @ y @ carrier ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
member_a @ x @ carrier ).
% assms(1)
thf(fact_2_xy__in__eachothers__nbt,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
| ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ) ) ).
% xy_in_eachothers_nbt
thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) ).
% \<open>card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation)\<close>
thf(fact_4_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_5_nbt__subset__carrier,axiom,
! [X: a] :
( ( member_a @ X @ carrier )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ carrier ) ) ).
% nbt_subset_carrier
thf(fact_6_rational__preference__axioms,axiom,
prefer719835986ence_a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
finite_finite_a @ carrier ).
% \<open>finite carrier\<close>
thf(fact_8_preference__axioms,axiom,
prefer1292084992ence_a @ carrier @ relation ).
% preference_axioms
thf(fact_9_trans__refl,axiom,
order_preorder_on_a @ carrier @ relation ).
% trans_refl
thf(fact_10_nbt__subset,axiom,
! [X: a,Y: a] :
( ( finite_finite_a @ carrier )
=> ( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( 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 ) ) ) ) ) ) ).
% nbt_subset
thf(fact_11_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_12_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_13_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_14_total,axiom,
total_on_a @ carrier @ relation ).
% total
thf(fact_15_fnt__carrier__fnt__nbt,axiom,
! [X2: a] :
( ( member_a @ X2 @ carrier )
=> ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% fnt_carrier_fnt_nbt
thf(fact_16_same__nbt__same__pref,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ carrier )
=> ( ( member_a @ Y @ carrier )
=> ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
& ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% same_nbt_same_pref
thf(fact_17_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_18_reflexivity,axiom,
refl_on_a @ carrier @ relation ).
% reflexivity
thf(fact_19_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_20_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_21_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_22_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_23_indifferent__imp__weak__pref_I1_J,axiom,
! [X: a,Y: 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 @ X @ Y ) @ relation ) ) ).
% indifferent_imp_weak_pref(1)
thf(fact_24_indifferent__imp__weak__pref_I2_J,axiom,
! [X: a,Y: 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 @ X ) @ relation ) ) ).
% indifferent_imp_weak_pref(2)
thf(fact_25_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_26_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_27_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_28_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_29_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_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,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 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ).
% rational_preference.indifferent_imp_weak_pref(2)
thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,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 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ).
% rational_preference.indifferent_imp_weak_pref(1)
thf(fact_32_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_33_preference__def,axiom,
( prefer1293349710ce_nat
= ( ^ [Carrier2: set_nat,Relation2: set_Pr1986765409at_nat] :
( ! [X3: nat,Y2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
=> ( member_nat @ X3 @ Carrier2 ) )
& ! [X3: nat,Y2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
=> ( member_nat @ Y2 @ Carrier2 ) )
& ( order_1150839801on_nat @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_34_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_35_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_36_preference_Ointro,axiom,
! [Relation: set_Pr1986765409at_nat,Carrier: set_nat] :
( ! [X4: nat,Y3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
=> ( member_nat @ X4 @ Carrier ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
=> ( member_nat @ Y3 @ Carrier ) )
=> ( ( order_1150839801on_nat @ Carrier @ Relation )
=> ( prefer1293349710ce_nat @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_37_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_38_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_39_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_40_preference_Onot__outside,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1293349710ce_nat @ Carrier @ Relation )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
=> ( member_nat @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_41_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_42_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_43_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_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_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( total_on_a @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_46_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_47_rational__preference_OcompleteD,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ( X != Y )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
| ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.completeD
thf(fact_48_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_49_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_50_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_51_rational__preference_Oweak__is__transitive,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( ( member_nat @ X @ Carrier )
& ( member_nat @ Y @ Carrier )
& ( member_nat @ Z @ Carrier ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation ) ) ) ) ) ).
% rational_preference.weak_is_transitive
thf(fact_52_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_53_rational__preference_Ofnt__carrier__fnt__rel,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( finite179568208od_a_a @ Carrier )
=> ( finite1664988688od_a_a @ Relation ) ) ) ).
% rational_preference.fnt_carrier_fnt_rel
thf(fact_54_rational__preference_Ofnt__carrier__fnt__rel,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( finite_finite_nat @ Carrier )
=> ( finite772653738at_nat @ Relation ) ) ) ).
% rational_preference.fnt_carrier_fnt_rel
thf(fact_55_rational__preference_Ofnt__carrier__fnt__rel,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( finite_finite_a @ Carrier )
=> ( finite179568208od_a_a @ Relation ) ) ) ).
% rational_preference.fnt_carrier_fnt_rel
thf(fact_56_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_57_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( ( member_nat @ X @ Carrier )
& ( member_nat @ Y @ Carrier ) )
=> ( ( ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
= ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
& ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_58_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_59_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_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_66_rational__preference_Ostrict__is__neg__transitive,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( ( member_nat @ X @ Carrier )
& ( member_nat @ Y @ Carrier )
& ( member_nat @ Z @ Carrier ) )
=> ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
& ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
=> ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation )
& ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ X ) @ Relation ) )
| ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
& ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% rational_preference.strict_is_neg_transitive
thf(fact_67_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_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_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
=> ( member_nat @ Y @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.worse_in_no_better
thf(fact_70_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_71_rational__preference_Osame__nbt__same__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 )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
& ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.same_nbt_same_pref
thf(fact_72_rational__preference_Osame__nbt__same__pref,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
& ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) )
= ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.same_nbt_same_pref
thf(fact_73_rational__preference_Osame__nbt__same__pref,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 )
=> ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
& ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% rational_preference.same_nbt_same_pref
thf(fact_74_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( finite179568208od_a_a @ Carrier )
=> ! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ Carrier )
=> ( finite179568208od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% rational_preference.fnt_carrier_fnt_nbt
thf(fact_75_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( finite_finite_nat @ Carrier )
=> ! [X2: nat] :
( ( member_nat @ X2 @ Carrier )
=> ( finite_finite_nat @ ( prefer1672638789an_nat @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% rational_preference.fnt_carrier_fnt_nbt
thf(fact_76_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( finite_finite_a @ Carrier )
=> ! [X2: a] :
( ( member_a @ X2 @ Carrier )
=> ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% rational_preference.fnt_carrier_fnt_nbt
thf(fact_77_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_78_rational__preference_Ono__better__than__nonepty,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_nat )
=> ( ( member_nat @ X @ Carrier )
=> ( ( prefer1672638789an_nat @ X @ Carrier @ Relation )
!= bot_bot_set_nat ) ) ) ) ).
% rational_preference.no_better_than_nonepty
thf(fact_79_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_80_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_81_rational__preference_Ono__better__thansubset__rel,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ).
% rational_preference.no_better_thansubset_rel
thf(fact_82_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_83_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_84_rational__preference_Onbt__subset,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 )
=> ( ( finite179568208od_a_a @ Carrier )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ( member449909584od_a_a @ Y @ Carrier )
=> ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
| ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% rational_preference.nbt_subset
thf(fact_85_rational__preference_Onbt__subset,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( finite_finite_nat @ Carrier )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
| ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% rational_preference.nbt_subset
thf(fact_86_rational__preference_Onbt__subset,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( finite_finite_a @ Carrier )
=> ( ( member_a @ X @ Carrier )
=> ( ( member_a @ Y @ Carrier )
=> ( ( 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 ) ) ) ) ) ) ) ).
% rational_preference.nbt_subset
thf(fact_87_rational__preference_Oxy__in__eachothers__nbt,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 )
=> ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
| ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% rational_preference.xy_in_eachothers_nbt
thf(fact_88_rational__preference_Oxy__in__eachothers__nbt,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
| ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% rational_preference.xy_in_eachothers_nbt
thf(fact_89_rational__preference_Oxy__in__eachothers__nbt,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 )
=> ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
| ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% rational_preference.xy_in_eachothers_nbt
thf(fact_90_rational__preference_Onbt__subset__carrier,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ X @ Carrier )
=> ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% rational_preference.nbt_subset_carrier
thf(fact_91_rational__preference_Onbt__subset__carrier,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ Carrier )
=> ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% rational_preference.nbt_subset_carrier
thf(fact_92_rational__preference_Onbt__subset__carrier,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( member_a @ X @ Carrier )
=> ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% rational_preference.nbt_subset_carrier
thf(fact_93_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_94_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_95_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_96_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_97_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_98_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_99_Pair__le,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_100_Pair__le,axiom,
! [A: set_a,B: nat,C: set_a,D: nat] :
( ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ A @ B ) @ ( produc515611863_a_nat @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_101_Pair__le,axiom,
! [A: nat,B: set_a,C: nat,D: set_a] :
( ( ord_le841160035_set_a @ ( produc1450889781_set_a @ A @ B ) @ ( produc1450889781_set_a @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_102_Pair__le,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_103_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_104_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_105_fnt__carrier__fnt__rel,axiom,
( ( finite_finite_a @ carrier )
=> ( finite179568208od_a_a @ relation ) ) ).
% fnt_carrier_fnt_rel
thf(fact_106_card__mono,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_107_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_108_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_109_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_110_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_111_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member449909584od_a_a @ X3 @ A2 ) )
= ( A2 = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_112_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_113_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_114_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_115_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_116_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_117_subsetI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A2 )
=> ( member449909584od_a_a @ X4 @ B2 ) )
=> ( ord_le1824328871od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_118_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_119_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_120_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_121_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_122_at__least__as__goodD,axiom,
! [Z: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
( ( member_nat @ Z @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_123_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_124_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_125_at__lst__asgd__ge,axiom,
! [X: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
( ( member_nat @ X @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_126_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_127_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
= ( A2 != bot_bo2131659635od_a_a ) ) ).
% ex_in_conv
thf(fact_128_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_129_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_130_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member449909584od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo2131659635od_a_a ) ) ).
% equals0I
thf(fact_131_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_132_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_133_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_134_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_135_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_136_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% emptyE
thf(fact_137_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_138_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_139_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( member449909584od_a_a @ X @ A2 )
=> ( member449909584od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_140_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_141_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_142_subsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( member449909584od_a_a @ C @ A2 )
=> ( member449909584od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_143_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_144_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_145_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_146_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_147_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_148_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_149_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_150_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_151_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_152_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_153_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_154_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_155_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_156_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_157_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_158_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_159_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_160_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_161_rational__preference_Opref__in__at__least__as,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
=> ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% rational_preference.pref_in_at_least_as
thf(fact_162_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_163_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_164_rational__preference_Oas__good__as__sameIff,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,Z: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ Z @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
= ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ).
% rational_preference.as_good_as_sameIff
thf(fact_165_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_166_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_167_rational__preference_Oas__good__asIff,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( member_nat @ X @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
= ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ).
% rational_preference.as_good_asIff
thf(fact_168_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_169_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_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( prefer1147844220ce_nat @ Carrier @ Relation )
=> ( ( Carrier != bot_bot_set_nat )
=> ( ( member_nat @ X @ Carrier )
=> ( ( member_nat @ Y @ Carrier )
=> ( ~ ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% rational_preference.at_lst_asgd_not_ge
thf(fact_171_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_172_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_173_Pair__mono,axiom,
! [X: set_a,X5: set_a,Y: nat,Y5: nat] :
( ( ord_less_eq_set_a @ X @ X5 )
=> ( ( ord_less_eq_nat @ Y @ Y5 )
=> ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ X @ Y ) @ ( produc515611863_a_nat @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_174_Pair__mono,axiom,
! [X: nat,X5: nat,Y: set_a,Y5: set_a] :
( ( ord_less_eq_nat @ X @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le841160035_set_a @ ( produc1450889781_set_a @ X @ Y ) @ ( produc1450889781_set_a @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_175_Pair__mono,axiom,
! [X: nat,X5: nat,Y: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ X5 )
=> ( ( ord_less_eq_nat @ Y @ Y5 )
=> ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_176_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_177_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_178_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_179_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_180_bot__prod__def,axiom,
( bot_bo1519632275_set_a
= ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% bot_prod_def
thf(fact_181_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_a_a] :
( ~ ( finite179568208od_a_a @ S )
=> ( S != bot_bo2131659635od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_182_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_183_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_184_finite_OemptyI,axiom,
finite179568208od_a_a @ bot_bo2131659635od_a_a ).
% finite.emptyI
thf(fact_185_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_186_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_187_finite__subset,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( finite179568208od_a_a @ B2 )
=> ( finite179568208od_a_a @ A2 ) ) ) ).
% finite_subset
thf(fact_188_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_189_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_190_infinite__super,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ S @ T2 )
=> ( ~ ( finite179568208od_a_a @ S )
=> ~ ( finite179568208od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_191_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_192_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_193_rev__finite__subset,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( finite179568208od_a_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_194_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_195_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_196_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_197_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_198_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_199_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_200_card__subset__eq,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( ( finite1481642319od_a_a @ A2 )
= ( finite1481642319od_a_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_201_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_202_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_203_infinite__arbitrarily__large,axiom,
! [A2: set_Product_prod_a_a,N: nat] :
( ~ ( finite179568208od_a_a @ A2 )
=> ? [B4: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B4 )
& ( ( finite1481642319od_a_a @ B4 )
= N )
& ( ord_le1824328871od_a_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_204_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_205_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ( finite_card_a @ B4 )
= N )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_206_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_Product_prod_a_a,C2: nat] :
( ! [G: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ G @ F )
=> ( ( finite179568208od_a_a @ G )
=> ( ord_less_eq_nat @ ( finite1481642319od_a_a @ G ) @ C2 ) ) )
=> ( ( finite179568208od_a_a @ F )
& ( ord_less_eq_nat @ ( finite1481642319od_a_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_207_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_208_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_209_card__seteq,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite179568208od_a_a @ B2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1481642319od_a_a @ B2 ) @ ( finite1481642319od_a_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_210_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_211_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_212_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ N @ ( finite1481642319od_a_a @ S ) )
=> ~ ! [T3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ T3 @ S )
=> ( ( ( finite1481642319od_a_a @ T3 )
= N )
=> ~ ( finite179568208od_a_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_213_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_214_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_215_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_Product_prod_a_a,R: product_prod_a_a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: product_prod_a_a] :
( ( member449909584od_a_a @ A4 @ A2 )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: a] :
( ( member449909584od_a_a @ A1 @ A2 )
=> ( ( member449909584od_a_a @ A22 @ A2 )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_216_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_217_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_218_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,R: product_prod_a_a > product_prod_a_a > $o] :
( ( finite179568208od_a_a @ B2 )
=> ( ! [A4: product_prod_a_a] :
( ( member449909584od_a_a @ A4 @ A2 )
=> ? [B5: product_prod_a_a] :
( ( member449909584od_a_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: product_prod_a_a] :
( ( member449909584od_a_a @ A1 @ A2 )
=> ( ( member449909584od_a_a @ A22 @ A2 )
=> ( ( member449909584od_a_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_219_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_prod_a_a,A2: set_nat,R: nat > product_prod_a_a > $o] :
( ( finite179568208od_a_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: product_prod_a_a] :
( ( member449909584od_a_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: product_prod_a_a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member449909584od_a_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_220_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_prod_a_a,A2: set_a,R: a > product_prod_a_a > $o] :
( ( finite179568208od_a_a @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B5: product_prod_a_a] :
( ( member449909584od_a_a @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: product_prod_a_a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member449909584od_a_a @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_221_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_Product_prod_a_a,R: product_prod_a_a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: product_prod_a_a] :
( ( member449909584od_a_a @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: nat] :
( ( member449909584od_a_a @ A1 @ A2 )
=> ( ( member449909584od_a_a @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_222_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_223_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_224_ex__card,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ A2 ) )
=> ? [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ A2 )
& ( ( finite_card_a @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_225_old_Oprod_Oinject,axiom,
! [A: a,B: a,A5: a,B7: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A5 @ B7 ) )
= ( ( A = A5 )
& ( B = B7 ) ) ) ).
% old.prod.inject
thf(fact_226_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_227_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_228_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_229_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_230_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_231_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_232_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A6: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_233_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A6: nat,B8: nat] :
( ( ord_less_eq_nat @ B8 @ A6 )
& ( ord_less_eq_nat @ A6 @ B8 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_234_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_235_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_236_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B6: nat] :
( ( ord_less_eq_nat @ A4 @ B6 )
=> ( P @ A4 @ B6 ) )
=> ( ! [A4: nat,B6: nat] :
( ( P @ B6 @ A4 )
=> ( P @ A4 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_237_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_238_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_239_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_240_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_241_order__class_Oorder_Oantisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_242_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_243_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_244_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_245_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_246_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : Y4 = Z2 )
= ( ^ [A6: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A6 @ B8 )
& ( ord_less_eq_set_a @ B8 @ A6 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A6: nat,B8: nat] :
( ( ord_less_eq_nat @ A6 @ B8 )
& ( ord_less_eq_nat @ B8 @ A6 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_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_250_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_251_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_252_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_253_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_254_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_255_eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% eq_refl
thf(fact_256_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_257_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_258_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_259_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_260_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_261_eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_262_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_263_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_264_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_265_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_266_ord__eq__le__subst,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F2 @ 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 @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_267_ord__eq__le__subst,axiom,
! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_268_ord__eq__le__subst,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_269_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_270_order__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_271_order__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_272_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_273_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_274_order__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F2 @ 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 @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_275_order__subst1,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_276_order__subst1,axiom,
! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_277_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_278_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X4: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_279_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_280_Pair__inject,axiom,
! [A: a,B: a,A5: a,B7: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A5 @ B7 ) )
=> ~ ( ( A = A5 )
=> ( B != B7 ) ) ) ).
% Pair_inject
thf(fact_281_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A4: a,B6: a] :
( Y
!= ( product_Pair_a_a @ A4 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_282_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_283_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_284_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_285_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_286_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_287_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_288_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_289_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_290_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_291_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% preorder_on_empty
thf(fact_292_total__onI,axiom,
! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A2 )
=> ( ( member449909584od_a_a @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_1490170027od_a_a @ A2 @ R ) ) ).
% total_onI
thf(fact_293_total__onI,axiom,
! [A2: set_nat,R: set_Pr1986765409at_nat] :
( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
| ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_on_nat @ A2 @ R ) ) ).
% total_onI
thf(fact_294_total__onI,axiom,
! [A2: set_a,R: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
=> ( total_on_a @ A2 @ R ) ) ).
% total_onI
thf(fact_295_total__on__def,axiom,
( total_on_a
= ( ^ [A3: set_a,R2: 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 ) @ R2 )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_296_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_297_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_298_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_299_subrelI,axiom,
! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le1824328871od_a_a @ R @ S3 ) ) ).
% subrelI
thf(fact_300_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% refl_on_empty
thf(fact_301_refl__on__domain,axiom,
! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A2 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
=> ( ( member449909584od_a_a @ A @ A2 )
& ( member449909584od_a_a @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_302_refl__on__domain,axiom,
! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat,B: nat] :
( ( refl_on_nat @ A2 @ R )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A2 )
& ( member_nat @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_303_refl__on__domain,axiom,
! [A2: set_a,R: set_Product_prod_a_a,A: a,B: a] :
( ( refl_on_a @ A2 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( ( member_a @ A @ A2 )
& ( member_a @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_304_refl__onD2,axiom,
! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A2 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
=> ( member449909584od_a_a @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_305_refl__onD2,axiom,
! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A2 @ R )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member_nat @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_306_refl__onD2,axiom,
! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A2 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member_a @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_307_refl__onD1,axiom,
! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A2 @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
=> ( member449909584od_a_a @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_308_refl__onD1,axiom,
! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A2 @ R )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member_nat @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_309_refl__onD1,axiom,
! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A2 @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member_a @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_310_refl__onD,axiom,
! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A2 @ R )
=> ( ( member449909584od_a_a @ A @ A2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_311_refl__onD,axiom,
! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat] :
( ( refl_on_nat @ A2 @ R )
=> ( ( member_nat @ A @ A2 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_312_refl__onD,axiom,
! [A2: set_a,R: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A2 @ R )
=> ( ( member_a @ A @ A2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_313_total__on__empty,axiom,
! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% total_on_empty
thf(fact_314_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_315_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_316_finite__transitivity__chain,axiom,
! [A2: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
( ( finite179568208od_a_a @ A2 )
=> ( ! [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 @ A2 )
=> ? [Y6: product_prod_a_a] :
( ( member449909584od_a_a @ Y6 @ A2 )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A2 = bot_bo2131659635od_a_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_317_finite__transitivity__chain,axiom,
! [A2: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
~ ( R3 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R3 @ X4 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_318_finite__transitivity__chain,axiom,
! [A2: set_a,R3: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [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 @ A2 )
=> ? [Y6: a] :
( ( member_a @ Y6 @ A2 )
& ( R3 @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_319_subset__emptyI,axiom,
! [A2: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member449909584od_a_a @ X4 @ A2 )
=> ( ord_le1824328871od_a_a @ A2 @ bot_bo2131659635od_a_a ) ) ).
% subset_emptyI
thf(fact_320_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_321_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X4: a] :
~ ( member_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_322_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_323_ssubst__Pair__rhs,axiom,
! [R: a,S3: a,R3: set_Product_prod_a_a,S4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S3 ) @ R3 )
=> ( ( S4 = S3 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_324_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_325_rational__preference__axioms__def,axiom,
prefer580340847ioms_a = total_on_a ).
% rational_preference_axioms_def
thf(fact_326_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_327_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_328_finite__indexed__bound,axiom,
! [A2: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ? [X_1: nat] : ( P @ X4 @ X_1 ) )
=> ? [M2: nat] :
! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M2 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_329_finite__indexed__bound,axiom,
! [A2: set_Product_prod_a_a,P: product_prod_a_a > nat > $o] :
( ( finite179568208od_a_a @ A2 )
=> ( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A2 )
=> ? [X_1: nat] : ( P @ X4 @ X_1 ) )
=> ? [M2: nat] :
! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M2 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_330_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [X_1: nat] : ( P @ X4 @ X_1 ) )
=> ? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M2 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_331_transitivity,axiom,
trans_a @ relation ).
% transitivity
thf(fact_332_trans__empty,axiom,
trans_a @ bot_bo2131659635od_a_a ).
% trans_empty
thf(fact_333_transD,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_334_transE,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_335_transI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X4: a,Y3: a,Z3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
=> ( trans_a @ R ) ) ).
% transI
thf(fact_336_trans__def,axiom,
( trans_a
= ( ^ [R2: set_Product_prod_a_a] :
! [X3: a,Y2: a,Z4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% trans_def
thf(fact_337_preference_Otransitivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( trans_a @ Relation ) ) ).
% preference.transitivity
thf(fact_338_preorder__on__def,axiom,
( order_preorder_on_a
= ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
( ( refl_on_a @ A3 @ R2 )
& ( trans_a @ R2 ) ) ) ) ).
% preorder_on_def
thf(fact_339_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : A3 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_340_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_341_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_342_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_343_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_344_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_345_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_346_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_347_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_348_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_349_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_350_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_351_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_352_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_353_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) ).
%------------------------------------------------------------------------------