TPTP Problem File: ITP205^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP205^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_966__6345404_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 : USubst/prob_966__6345404_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 419 ( 232 unt; 64 typ; 0 def)
% Number of atoms : 868 ( 399 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2595 ( 72 ~; 9 |; 47 &;2211 @)
% ( 0 <=>; 256 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 54 usr; 5 con; 0-4 aty)
% Number of variables : 982 ( 157 ^; 808 !; 17 ?; 982 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:23.013
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_J,type,
produc1418842292n_game: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
set_Pr1625152599n_game: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
set_Pr166476775n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
produc1016592119n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
produc1078154247n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
produc735959047riable: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Ogame_J,type,
option_game: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Ovariable_J,type,
set_variable: $tType ).
thf(ty_n_t__Syntax__Ovariable,type,
variable: $tType ).
thf(ty_n_t__Syntax__Ogame,type,
game: $tType ).
% Explicit typings (54)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
bNF_Ca1986151756riable: set_Pr1625152599n_game > ( option_game > set_variable ) > $o ).
thf(sy_c_BNF__Def_OfstOp_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
bNF_fs984229733n_game: ( set_variable > option_game > $o ) > ( option_game > option_game > $o ) > produc1078154247n_game > produc1078154247n_game ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
bNF_pi856525207n_game: ( set_variable > option_game > $o ) > ( option_game > option_game > $o ) > set_variable > option_game > option_game ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
bNF_pi1813424679n_game: ( set_variable > set_variable > $o ) > ( set_variable > option_game > $o ) > set_variable > option_game > set_variable ).
thf(sy_c_BNF__Def_OsndOp_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
bNF_sn1374695591n_game: ( set_variable > option_game > $o ) > ( option_game > option_game > $o ) > produc1078154247n_game > produc1016592119n_game ).
thf(sy_c_BNF__Def_OsndOp_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
bNF_sn582998327n_game: ( set_variable > set_variable > $o ) > ( set_variable > option_game > $o ) > produc1078154247n_game > produc1078154247n_game ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Option__Ooption_It__Syntax__Ogame_J_M_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J_J,type,
sup_su169929796game_o: ( option_game > option_game > $o ) > ( option_game > option_game > $o ) > option_game > option_game > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J,type,
sup_su1229228248game_o: ( option_game > $o ) > ( option_game > $o ) > option_game > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Syntax__Ovariable_J_M_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J_J,type,
sup_su1227468340game_o: ( set_variable > option_game > $o ) > ( set_variable > option_game > $o ) > set_variable > option_game > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Syntax__Ovariable_M_Eo_J,type,
sup_sup_variable_o: ( variable > $o ) > ( variable > $o ) > variable > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
sup_su423136299n_game: set_Pr1625152599n_game > set_Pr1625152599n_game > set_Pr1625152599n_game ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
sup_su796825019n_game: set_Pr166476775n_game > set_Pr166476775n_game > set_Pr166476775n_game ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Syntax__Ovariable_J,type,
sup_sup_set_variable: set_variable > set_variable > set_variable ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ogame,type,
none_game: option_game ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ogame,type,
some_game: game > option_game ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Ogame,type,
the_game: option_game > game ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Option__Ooption_It__Syntax__Ogame_J_M_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J_J,type,
ord_le707292176game_o: ( option_game > option_game > $o ) > ( option_game > option_game > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Syntax__Ovariable_J_M_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J_J,type,
ord_le2134856704game_o: ( set_variable > option_game > $o ) > ( set_variable > option_game > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Ovariable_M_Eo_J,type,
ord_le1407353162able_o: ( variable > $o ) > ( variable > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
ord_le1780499447n_game: set_Pr1625152599n_game > set_Pr1625152599n_game > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
ord_le17855367n_game: set_Pr166476775n_game > set_Pr166476775n_game > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ovariable_J,type,
ord_le282106107riable: set_variable > set_variable > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Syntax__Ovariable_J,type,
order_272405634riable: ( set_variable > $o ) > set_variable ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1111617711n_game: option_game > option_game > produc1016592119n_game ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc1140431679riable: option_game > set_variable > produc735959047riable ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1149443391n_game: set_variable > option_game > produc1078154247n_game ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1942569115n_game: produc1016592119n_game > option_game ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc884810027riable: produc735959047riable > option_game ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc893821739n_game: produc1078154247n_game > set_variable ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc279011037n_game: produc1016592119n_game > option_game ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc284475501riable: produc735959047riable > set_variable ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc293487213n_game: produc1078154247n_game > option_game ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1696194127n_game: produc1016592119n_game > produc1016592119n_game ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc345397471riable: produc735959047riable > produc1078154247n_game ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc354409183n_game: produc1078154247n_game > produc735959047riable ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
collec884051426n_game: ( produc1016592119n_game > $o ) > set_Pr1625152599n_game ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
collec1702522994n_game: ( produc1078154247n_game > $o ) > set_Pr166476775n_game ).
thf(sy_c_Set_OCollect_001t__Syntax__Ovariable,type,
collect_variable: ( variable > $o ) > set_variable ).
thf(sy_c_Static__Semantics_OBVG,type,
static_BVG: game > set_variable ).
thf(sy_c_Syntax_Ogame_OChoice,type,
choice: game > game > game ).
thf(sy_c_Syntax_Ogame_OCompose,type,
compose: game > game > game ).
thf(sy_c_Syntax_Ogame_OLoop,type,
loop: game > game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OChoiceo,type,
uSubst1484167963hoiceo: option_game > option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OComposeo,type,
uSubst532817840mposeo: option_game > option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OLoopo,type,
uSubst23177304_Loopo: option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousubstappp,type,
uSubst516392814stappp: produc1418842292n_game > set_variable > game > produc1078154247n_game ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
member191809696n_game: produc1016592119n_game > set_Pr1625152599n_game > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
member171223600n_game: produc1078154247n_game > set_Pr166476775n_game > $o ).
thf(sy_c_member_001t__Syntax__Ovariable,type,
member_variable: variable > set_variable > $o ).
thf(sy_v_Ua____,type,
ua: set_variable ).
thf(sy_v__092_060alpha_062_H____,type,
alpha: game ).
thf(sy_v__092_060beta_062____,type,
beta: game ).
thf(sy_v__092_060sigma_062,type,
sigma: produc1418842292n_game ).
% Relevant facts (354)
thf(fact_0__092_060open_062U_A_092_060union_062_ABVG_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_A_092_060union_062_ABVG_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_A_Ifst_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_A_092_060beta_062_J_J_J_A_092_060subseteq_062_Afst_A_Iusubstappp_A_092_060sigma_062_A_Ifst_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_A_092_060beta_062_J_092_060close_062,axiom,
ord_le282106107riable @ ( sup_sup_set_variable @ ( sup_sup_set_variable @ ua @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) ) ) @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
% \<open>U \<union> BVG (the (snd (usubstappp \<sigma> U \<alpha>))) \<union> BVG (the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>))) \<subseteq> fst (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>)\<close>
thf(fact_1_fact,axiom,
ord_le282106107riable @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) ) ) ) @ ( sup_sup_set_variable @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) ) @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) ).
% fact
thf(fact_2_IHa,axiom,
ord_le282106107riable @ ( sup_sup_set_variable @ ua @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ).
% IHa
thf(fact_3_IHb,axiom,
ord_le282106107riable @ ( sup_sup_set_variable @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
% IHb
thf(fact_4_Compose_Oprems,axiom,
( ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) )
!= none_game ) ).
% Compose.prems
thf(fact_5_usubst__taboos__mon,axiom,
! [U: set_variable,Sigma: produc1418842292n_game,Alpha: game] : ( ord_le282106107riable @ U @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) ) ).
% usubst_taboos_mon
thf(fact_6_usubstappp__fst__mon,axiom,
! [U: set_variable,V: set_variable,Sigma: produc1418842292n_game,Alpha: game] :
( ( ord_le282106107riable @ U @ V )
=> ( ord_le282106107riable @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ).
% usubstappp_fst_mon
thf(fact_7_Compose_OIH_I2_J,axiom,
! [U: set_variable] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ U @ beta ) )
!= none_game )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ U @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ U @ beta ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ U @ beta ) ) ) ) ).
% Compose.IH(2)
thf(fact_8_Compose_OIH_I1_J,axiom,
! [U: set_variable] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ U @ alpha ) )
!= none_game )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ U @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ U @ alpha ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ U @ alpha ) ) ) ) ).
% Compose.IH(1)
thf(fact_9_Un__subset__iff,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game,C: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ ( sup_su423136299n_game @ A @ B ) @ C )
= ( ( ord_le1780499447n_game @ A @ C )
& ( ord_le1780499447n_game @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_10_Un__subset__iff,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game,C: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ ( sup_su796825019n_game @ A @ B ) @ C )
= ( ( ord_le17855367n_game @ A @ C )
& ( ord_le17855367n_game @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_11_Un__subset__iff,axiom,
! [A: set_variable,B: set_variable,C: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ C )
= ( ( ord_le282106107riable @ A @ C )
& ( ord_le282106107riable @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_12_le__sup__iff,axiom,
! [X: variable > $o,Y: variable > $o,Z: variable > $o] :
( ( ord_le1407353162able_o @ ( sup_sup_variable_o @ X @ Y ) @ Z )
= ( ( ord_le1407353162able_o @ X @ Z )
& ( ord_le1407353162able_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_13_le__sup__iff,axiom,
! [X: set_Pr1625152599n_game,Y: set_Pr1625152599n_game,Z: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ ( sup_su423136299n_game @ X @ Y ) @ Z )
= ( ( ord_le1780499447n_game @ X @ Z )
& ( ord_le1780499447n_game @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_14_le__sup__iff,axiom,
! [X: set_Pr166476775n_game,Y: set_Pr166476775n_game,Z: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ ( sup_su796825019n_game @ X @ Y ) @ Z )
= ( ( ord_le17855367n_game @ X @ Z )
& ( ord_le17855367n_game @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_15_le__sup__iff,axiom,
! [X: option_game > option_game > $o,Y: option_game > option_game > $o,Z: option_game > option_game > $o] :
( ( ord_le707292176game_o @ ( sup_su169929796game_o @ X @ Y ) @ Z )
= ( ( ord_le707292176game_o @ X @ Z )
& ( ord_le707292176game_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_16_le__sup__iff,axiom,
! [X: set_variable > option_game > $o,Y: set_variable > option_game > $o,Z: set_variable > option_game > $o] :
( ( ord_le2134856704game_o @ ( sup_su1227468340game_o @ X @ Y ) @ Z )
= ( ( ord_le2134856704game_o @ X @ Z )
& ( ord_le2134856704game_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_17_le__sup__iff,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( ( ord_le282106107riable @ X @ Z )
& ( ord_le282106107riable @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_18_sup_Obounded__iff,axiom,
! [B2: variable > $o,C2: variable > $o,A2: variable > $o] :
( ( ord_le1407353162able_o @ ( sup_sup_variable_o @ B2 @ C2 ) @ A2 )
= ( ( ord_le1407353162able_o @ B2 @ A2 )
& ( ord_le1407353162able_o @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_19_sup_Obounded__iff,axiom,
! [B2: set_Pr1625152599n_game,C2: set_Pr1625152599n_game,A2: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ ( sup_su423136299n_game @ B2 @ C2 ) @ A2 )
= ( ( ord_le1780499447n_game @ B2 @ A2 )
& ( ord_le1780499447n_game @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_20_sup_Obounded__iff,axiom,
! [B2: set_Pr166476775n_game,C2: set_Pr166476775n_game,A2: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ ( sup_su796825019n_game @ B2 @ C2 ) @ A2 )
= ( ( ord_le17855367n_game @ B2 @ A2 )
& ( ord_le17855367n_game @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_21_sup_Obounded__iff,axiom,
! [B2: option_game > option_game > $o,C2: option_game > option_game > $o,A2: option_game > option_game > $o] :
( ( ord_le707292176game_o @ ( sup_su169929796game_o @ B2 @ C2 ) @ A2 )
= ( ( ord_le707292176game_o @ B2 @ A2 )
& ( ord_le707292176game_o @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_22_sup_Obounded__iff,axiom,
! [B2: set_variable > option_game > $o,C2: set_variable > option_game > $o,A2: set_variable > option_game > $o] :
( ( ord_le2134856704game_o @ ( sup_su1227468340game_o @ B2 @ C2 ) @ A2 )
= ( ( ord_le2134856704game_o @ B2 @ A2 )
& ( ord_le2134856704game_o @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_23_sup_Obounded__iff,axiom,
! [B2: set_variable,C2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C2 ) @ A2 )
= ( ( ord_le282106107riable @ B2 @ A2 )
& ( ord_le282106107riable @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_24_BVG__compose,axiom,
! [Alpha: game,Beta: game] : ( ord_le282106107riable @ ( static_BVG @ ( compose @ Alpha @ Beta ) ) @ ( sup_sup_set_variable @ ( static_BVG @ Alpha ) @ ( static_BVG @ Beta ) ) ) ).
% BVG_compose
thf(fact_25_game_Oinject_I5_J,axiom,
! [X51: game,X52: game,Y51: game,Y52: game] :
( ( ( compose @ X51 @ X52 )
= ( compose @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% game.inject(5)
thf(fact_26_UnCI,axiom,
! [C2: produc1016592119n_game,B: set_Pr1625152599n_game,A: set_Pr1625152599n_game] :
( ( ~ ( member191809696n_game @ C2 @ B )
=> ( member191809696n_game @ C2 @ A ) )
=> ( member191809696n_game @ C2 @ ( sup_su423136299n_game @ A @ B ) ) ) ).
% UnCI
thf(fact_27_UnCI,axiom,
! [C2: produc1078154247n_game,B: set_Pr166476775n_game,A: set_Pr166476775n_game] :
( ( ~ ( member171223600n_game @ C2 @ B )
=> ( member171223600n_game @ C2 @ A ) )
=> ( member171223600n_game @ C2 @ ( sup_su796825019n_game @ A @ B ) ) ) ).
% UnCI
thf(fact_28_UnCI,axiom,
! [C2: variable,B: set_variable,A: set_variable] :
( ( ~ ( member_variable @ C2 @ B )
=> ( member_variable @ C2 @ A ) )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnCI
thf(fact_29_union__or,axiom,
! [C2: produc1016592119n_game,A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ( member191809696n_game @ C2 @ ( sup_su423136299n_game @ A @ B ) )
= ( ( member191809696n_game @ C2 @ A )
| ( member191809696n_game @ C2 @ B ) ) ) ).
% union_or
thf(fact_30_union__or,axiom,
! [C2: produc1078154247n_game,A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ( member171223600n_game @ C2 @ ( sup_su796825019n_game @ A @ B ) )
= ( ( member171223600n_game @ C2 @ A )
| ( member171223600n_game @ C2 @ B ) ) ) ).
% union_or
thf(fact_31_union__or,axiom,
! [C2: variable,A: set_variable,B: set_variable] :
( ( member_variable @ C2 @ ( sup_sup_set_variable @ A @ B ) )
= ( ( member_variable @ C2 @ A )
| ( member_variable @ C2 @ B ) ) ) ).
% union_or
thf(fact_32_sup__apply,axiom,
( sup_sup_variable_o
= ( ^ [F: variable > $o,G: variable > $o,X2: variable] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_apply
thf(fact_33_sup__apply,axiom,
( sup_su169929796game_o
= ( ^ [F: option_game > option_game > $o,G: option_game > option_game > $o,X2: option_game] : ( sup_su1229228248game_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_apply
thf(fact_34_sup__apply,axiom,
( sup_su1227468340game_o
= ( ^ [F: set_variable > option_game > $o,G: set_variable > option_game > $o,X2: set_variable] : ( sup_su1229228248game_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_apply
thf(fact_35_sup_Oidem,axiom,
! [A2: variable > $o] :
( ( sup_sup_variable_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_36_sup_Oidem,axiom,
! [A2: option_game > option_game > $o] :
( ( sup_su169929796game_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_37_sup_Oidem,axiom,
! [A2: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_38_sup_Oidem,axiom,
! [A2: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_39_sup_Oidem,axiom,
! [A2: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_40_sup_Oidem,axiom,
! [A2: set_variable] :
( ( sup_sup_set_variable @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_41_sup__idem,axiom,
! [X: variable > $o] :
( ( sup_sup_variable_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_42_sup__idem,axiom,
! [X: option_game > option_game > $o] :
( ( sup_su169929796game_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_43_sup__idem,axiom,
! [X: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ X @ X )
= X ) ).
% sup_idem
thf(fact_44_sup__idem,axiom,
! [X: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_45_sup__idem,axiom,
! [X: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ X @ X )
= X ) ).
% sup_idem
thf(fact_46_sup__idem,axiom,
! [X: set_variable] :
( ( sup_sup_set_variable @ X @ X )
= X ) ).
% sup_idem
thf(fact_47_subset__antisym,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ A @ B )
=> ( ( ord_le1780499447n_game @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_48_subset__antisym,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ A @ B )
=> ( ( ord_le17855367n_game @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_49_subset__antisym,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_50_subsetI,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ! [X3: produc1016592119n_game] :
( ( member191809696n_game @ X3 @ A )
=> ( member191809696n_game @ X3 @ B ) )
=> ( ord_le1780499447n_game @ A @ B ) ) ).
% subsetI
thf(fact_51_subsetI,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ! [X3: produc1078154247n_game] :
( ( member171223600n_game @ X3 @ A )
=> ( member171223600n_game @ X3 @ B ) )
=> ( ord_le17855367n_game @ A @ B ) ) ).
% subsetI
thf(fact_52_subsetI,axiom,
! [A: set_variable,B: set_variable] :
( ! [X3: variable] :
( ( member_variable @ X3 @ A )
=> ( member_variable @ X3 @ B ) )
=> ( ord_le282106107riable @ A @ B ) ) ).
% subsetI
thf(fact_53_sup_Oright__idem,axiom,
! [A2: variable > $o,B2: variable > $o] :
( ( sup_sup_variable_o @ ( sup_sup_variable_o @ A2 @ B2 ) @ B2 )
= ( sup_sup_variable_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_54_sup_Oright__idem,axiom,
! [A2: option_game > option_game > $o,B2: option_game > option_game > $o] :
( ( sup_su169929796game_o @ ( sup_su169929796game_o @ A2 @ B2 ) @ B2 )
= ( sup_su169929796game_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_55_sup_Oright__idem,axiom,
! [A2: set_Pr1625152599n_game,B2: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ ( sup_su423136299n_game @ A2 @ B2 ) @ B2 )
= ( sup_su423136299n_game @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_56_sup_Oright__idem,axiom,
! [A2: set_variable > option_game > $o,B2: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ ( sup_su1227468340game_o @ A2 @ B2 ) @ B2 )
= ( sup_su1227468340game_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_57_sup_Oright__idem,axiom,
! [A2: set_Pr166476775n_game,B2: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ ( sup_su796825019n_game @ A2 @ B2 ) @ B2 )
= ( sup_su796825019n_game @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_58_sup_Oright__idem,axiom,
! [A2: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_variable @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_59_sup__left__idem,axiom,
! [X: variable > $o,Y: variable > $o] :
( ( sup_sup_variable_o @ X @ ( sup_sup_variable_o @ X @ Y ) )
= ( sup_sup_variable_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_60_sup__left__idem,axiom,
! [X: option_game > option_game > $o,Y: option_game > option_game > $o] :
( ( sup_su169929796game_o @ X @ ( sup_su169929796game_o @ X @ Y ) )
= ( sup_su169929796game_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_61_sup__left__idem,axiom,
! [X: set_Pr1625152599n_game,Y: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ X @ ( sup_su423136299n_game @ X @ Y ) )
= ( sup_su423136299n_game @ X @ Y ) ) ).
% sup_left_idem
thf(fact_62_sup__left__idem,axiom,
! [X: set_variable > option_game > $o,Y: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ X @ ( sup_su1227468340game_o @ X @ Y ) )
= ( sup_su1227468340game_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_63_sup__left__idem,axiom,
! [X: set_Pr166476775n_game,Y: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ X @ ( sup_su796825019n_game @ X @ Y ) )
= ( sup_su796825019n_game @ X @ Y ) ) ).
% sup_left_idem
thf(fact_64_sup__left__idem,axiom,
! [X: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ X @ Y ) )
= ( sup_sup_set_variable @ X @ Y ) ) ).
% sup_left_idem
thf(fact_65_sup_Oleft__idem,axiom,
! [A2: variable > $o,B2: variable > $o] :
( ( sup_sup_variable_o @ A2 @ ( sup_sup_variable_o @ A2 @ B2 ) )
= ( sup_sup_variable_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_66_sup_Oleft__idem,axiom,
! [A2: option_game > option_game > $o,B2: option_game > option_game > $o] :
( ( sup_su169929796game_o @ A2 @ ( sup_su169929796game_o @ A2 @ B2 ) )
= ( sup_su169929796game_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_67_sup_Oleft__idem,axiom,
! [A2: set_Pr1625152599n_game,B2: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ A2 @ ( sup_su423136299n_game @ A2 @ B2 ) )
= ( sup_su423136299n_game @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_68_sup_Oleft__idem,axiom,
! [A2: set_variable > option_game > $o,B2: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ A2 @ ( sup_su1227468340game_o @ A2 @ B2 ) )
= ( sup_su1227468340game_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_69_sup_Oleft__idem,axiom,
! [A2: set_Pr166476775n_game,B2: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ A2 @ ( sup_su796825019n_game @ A2 @ B2 ) )
= ( sup_su796825019n_game @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_70_sup_Oleft__idem,axiom,
! [A2: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ A2 @ B2 ) )
= ( sup_sup_set_variable @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_71_usubstappp__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,V: set_variable] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) )
!= none_game )
=> ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
= ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_det
thf(fact_72_Collect__mono__iff,axiom,
! [P: produc1016592119n_game > $o,Q: produc1016592119n_game > $o] :
( ( ord_le1780499447n_game @ ( collec884051426n_game @ P ) @ ( collec884051426n_game @ Q ) )
= ( ! [X2: produc1016592119n_game] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_73_Collect__mono__iff,axiom,
! [P: produc1078154247n_game > $o,Q: produc1078154247n_game > $o] :
( ( ord_le17855367n_game @ ( collec1702522994n_game @ P ) @ ( collec1702522994n_game @ Q ) )
= ( ! [X2: produc1078154247n_game] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_74_Collect__mono__iff,axiom,
! [P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) )
= ( ! [X2: variable] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_75_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr1625152599n_game,Z2: set_Pr1625152599n_game] : Y2 = Z2 )
= ( ^ [A3: set_Pr1625152599n_game,B3: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ A3 @ B3 )
& ( ord_le1780499447n_game @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_76_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr166476775n_game,Z2: set_Pr166476775n_game] : Y2 = Z2 )
= ( ^ [A3: set_Pr166476775n_game,B3: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ A3 @ B3 )
& ( ord_le17855367n_game @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_77_set__eq__subset,axiom,
( ( ^ [Y2: set_variable,Z2: set_variable] : Y2 = Z2 )
= ( ^ [A3: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
& ( ord_le282106107riable @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_78_subset__trans,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game,C: set_Pr1625152599n_game] :
( ( ord_le1780499447n_game @ A @ B )
=> ( ( ord_le1780499447n_game @ B @ C )
=> ( ord_le1780499447n_game @ A @ C ) ) ) ).
% subset_trans
thf(fact_79_subset__trans,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game,C: set_Pr166476775n_game] :
( ( ord_le17855367n_game @ A @ B )
=> ( ( ord_le17855367n_game @ B @ C )
=> ( ord_le17855367n_game @ A @ C ) ) ) ).
% subset_trans
thf(fact_80_subset__trans,axiom,
! [A: set_variable,B: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ C )
=> ( ord_le282106107riable @ A @ C ) ) ) ).
% subset_trans
thf(fact_81_Collect__mono,axiom,
! [P: produc1016592119n_game > $o,Q: produc1016592119n_game > $o] :
( ! [X3: produc1016592119n_game] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1780499447n_game @ ( collec884051426n_game @ P ) @ ( collec884051426n_game @ Q ) ) ) ).
% Collect_mono
thf(fact_82_Collect__mono,axiom,
! [P: produc1078154247n_game > $o,Q: produc1078154247n_game > $o] :
( ! [X3: produc1078154247n_game] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le17855367n_game @ ( collec1702522994n_game @ P ) @ ( collec1702522994n_game @ Q ) ) ) ).
% Collect_mono
thf(fact_83_Collect__mono,axiom,
! [P: variable > $o,Q: variable > $o] :
( ! [X3: variable] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) ) ) ).
% Collect_mono
thf(fact_84_subset__refl,axiom,
! [A: set_Pr1625152599n_game] : ( ord_le1780499447n_game @ A @ A ) ).
% subset_refl
thf(fact_85_subset__refl,axiom,
! [A: set_Pr166476775n_game] : ( ord_le17855367n_game @ A @ A ) ).
% subset_refl
thf(fact_86_subset__refl,axiom,
! [A: set_variable] : ( ord_le282106107riable @ A @ A ) ).
% subset_refl
thf(fact_87_subset__iff,axiom,
( ord_le1780499447n_game
= ( ^ [A3: set_Pr1625152599n_game,B3: set_Pr1625152599n_game] :
! [T: produc1016592119n_game] :
( ( member191809696n_game @ T @ A3 )
=> ( member191809696n_game @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_88_subset__iff,axiom,
( ord_le17855367n_game
= ( ^ [A3: set_Pr166476775n_game,B3: set_Pr166476775n_game] :
! [T: produc1078154247n_game] :
( ( member171223600n_game @ T @ A3 )
=> ( member171223600n_game @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_89_subset__iff,axiom,
( ord_le282106107riable
= ( ^ [A3: set_variable,B3: set_variable] :
! [T: variable] :
( ( member_variable @ T @ A3 )
=> ( member_variable @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_90_equalityD2,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ( A = B )
=> ( ord_le1780499447n_game @ B @ A ) ) ).
% equalityD2
thf(fact_91_equalityD2,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ( A = B )
=> ( ord_le17855367n_game @ B @ A ) ) ).
% equalityD2
thf(fact_92_equalityD2,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ B @ A ) ) ).
% equalityD2
thf(fact_93_equalityD1,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ( A = B )
=> ( ord_le1780499447n_game @ A @ B ) ) ).
% equalityD1
thf(fact_94_equalityD1,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ( A = B )
=> ( ord_le17855367n_game @ A @ B ) ) ).
% equalityD1
thf(fact_95_equalityD1,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ A @ B ) ) ).
% equalityD1
thf(fact_96_subset__eq,axiom,
( ord_le1780499447n_game
= ( ^ [A3: set_Pr1625152599n_game,B3: set_Pr1625152599n_game] :
! [X2: produc1016592119n_game] :
( ( member191809696n_game @ X2 @ A3 )
=> ( member191809696n_game @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_97_subset__eq,axiom,
( ord_le17855367n_game
= ( ^ [A3: set_Pr166476775n_game,B3: set_Pr166476775n_game] :
! [X2: produc1078154247n_game] :
( ( member171223600n_game @ X2 @ A3 )
=> ( member171223600n_game @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_98_subset__eq,axiom,
( ord_le282106107riable
= ( ^ [A3: set_variable,B3: set_variable] :
! [X2: variable] :
( ( member_variable @ X2 @ A3 )
=> ( member_variable @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_99_equalityE,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game] :
( ( A = B )
=> ~ ( ( ord_le1780499447n_game @ A @ B )
=> ~ ( ord_le1780499447n_game @ B @ A ) ) ) ).
% equalityE
thf(fact_100_equalityE,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game] :
( ( A = B )
=> ~ ( ( ord_le17855367n_game @ A @ B )
=> ~ ( ord_le17855367n_game @ B @ A ) ) ) ).
% equalityE
thf(fact_101_equalityE,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ~ ( ( ord_le282106107riable @ A @ B )
=> ~ ( ord_le282106107riable @ B @ A ) ) ) ).
% equalityE
thf(fact_102_subsetD,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game,C2: produc1016592119n_game] :
( ( ord_le1780499447n_game @ A @ B )
=> ( ( member191809696n_game @ C2 @ A )
=> ( member191809696n_game @ C2 @ B ) ) ) ).
% subsetD
thf(fact_103_subsetD,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game,C2: produc1078154247n_game] :
( ( ord_le17855367n_game @ A @ B )
=> ( ( member171223600n_game @ C2 @ A )
=> ( member171223600n_game @ C2 @ B ) ) ) ).
% subsetD
thf(fact_104_subsetD,axiom,
! [A: set_variable,B: set_variable,C2: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ C2 @ A )
=> ( member_variable @ C2 @ B ) ) ) ).
% subsetD
thf(fact_105_in__mono,axiom,
! [A: set_Pr1625152599n_game,B: set_Pr1625152599n_game,X: produc1016592119n_game] :
( ( ord_le1780499447n_game @ A @ B )
=> ( ( member191809696n_game @ X @ A )
=> ( member191809696n_game @ X @ B ) ) ) ).
% in_mono
thf(fact_106_in__mono,axiom,
! [A: set_Pr166476775n_game,B: set_Pr166476775n_game,X: produc1078154247n_game] :
( ( ord_le17855367n_game @ A @ B )
=> ( ( member171223600n_game @ X @ A )
=> ( member171223600n_game @ X @ B ) ) ) ).
% in_mono
thf(fact_107_in__mono,axiom,
! [A: set_variable,B: set_variable,X: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ X @ A )
=> ( member_variable @ X @ B ) ) ) ).
% in_mono
thf(fact_108_sup__left__commute,axiom,
! [X: variable > $o,Y: variable > $o,Z: variable > $o] :
( ( sup_sup_variable_o @ X @ ( sup_sup_variable_o @ Y @ Z ) )
= ( sup_sup_variable_o @ Y @ ( sup_sup_variable_o @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_109_sup__left__commute,axiom,
! [X: option_game > option_game > $o,Y: option_game > option_game > $o,Z: option_game > option_game > $o] :
( ( sup_su169929796game_o @ X @ ( sup_su169929796game_o @ Y @ Z ) )
= ( sup_su169929796game_o @ Y @ ( sup_su169929796game_o @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_110_sup__left__commute,axiom,
! [X: set_Pr1625152599n_game,Y: set_Pr1625152599n_game,Z: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ X @ ( sup_su423136299n_game @ Y @ Z ) )
= ( sup_su423136299n_game @ Y @ ( sup_su423136299n_game @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_111_sup__left__commute,axiom,
! [X: set_variable > option_game > $o,Y: set_variable > option_game > $o,Z: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ X @ ( sup_su1227468340game_o @ Y @ Z ) )
= ( sup_su1227468340game_o @ Y @ ( sup_su1227468340game_o @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_112_sup__left__commute,axiom,
! [X: set_Pr166476775n_game,Y: set_Pr166476775n_game,Z: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ X @ ( sup_su796825019n_game @ Y @ Z ) )
= ( sup_su796825019n_game @ Y @ ( sup_su796825019n_game @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_113_sup__left__commute,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) )
= ( sup_sup_set_variable @ Y @ ( sup_sup_set_variable @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_114_sup_Oleft__commute,axiom,
! [B2: variable > $o,A2: variable > $o,C2: variable > $o] :
( ( sup_sup_variable_o @ B2 @ ( sup_sup_variable_o @ A2 @ C2 ) )
= ( sup_sup_variable_o @ A2 @ ( sup_sup_variable_o @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_115_sup_Oleft__commute,axiom,
! [B2: option_game > option_game > $o,A2: option_game > option_game > $o,C2: option_game > option_game > $o] :
( ( sup_su169929796game_o @ B2 @ ( sup_su169929796game_o @ A2 @ C2 ) )
= ( sup_su169929796game_o @ A2 @ ( sup_su169929796game_o @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_116_sup_Oleft__commute,axiom,
! [B2: set_Pr1625152599n_game,A2: set_Pr1625152599n_game,C2: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ B2 @ ( sup_su423136299n_game @ A2 @ C2 ) )
= ( sup_su423136299n_game @ A2 @ ( sup_su423136299n_game @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_117_sup_Oleft__commute,axiom,
! [B2: set_variable > option_game > $o,A2: set_variable > option_game > $o,C2: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ B2 @ ( sup_su1227468340game_o @ A2 @ C2 ) )
= ( sup_su1227468340game_o @ A2 @ ( sup_su1227468340game_o @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_118_sup_Oleft__commute,axiom,
! [B2: set_Pr166476775n_game,A2: set_Pr166476775n_game,C2: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ B2 @ ( sup_su796825019n_game @ A2 @ C2 ) )
= ( sup_su796825019n_game @ A2 @ ( sup_su796825019n_game @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_119_sup_Oleft__commute,axiom,
! [B2: set_variable,A2: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ B2 @ ( sup_sup_set_variable @ A2 @ C2 ) )
= ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_120_sup__commute,axiom,
( sup_sup_variable_o
= ( ^ [X2: variable > $o,Y3: variable > $o] : ( sup_sup_variable_o @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_121_sup__commute,axiom,
( sup_su169929796game_o
= ( ^ [X2: option_game > option_game > $o,Y3: option_game > option_game > $o] : ( sup_su169929796game_o @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_122_sup__commute,axiom,
( sup_su423136299n_game
= ( ^ [X2: set_Pr1625152599n_game,Y3: set_Pr1625152599n_game] : ( sup_su423136299n_game @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_123_sup__commute,axiom,
( sup_su1227468340game_o
= ( ^ [X2: set_variable > option_game > $o,Y3: set_variable > option_game > $o] : ( sup_su1227468340game_o @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_124_sup__commute,axiom,
( sup_su796825019n_game
= ( ^ [X2: set_Pr166476775n_game,Y3: set_Pr166476775n_game] : ( sup_su796825019n_game @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_125_sup__commute,axiom,
( sup_sup_set_variable
= ( ^ [X2: set_variable,Y3: set_variable] : ( sup_sup_set_variable @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_126_sup_Ocommute,axiom,
( sup_sup_variable_o
= ( ^ [A4: variable > $o,B4: variable > $o] : ( sup_sup_variable_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_127_sup_Ocommute,axiom,
( sup_su169929796game_o
= ( ^ [A4: option_game > option_game > $o,B4: option_game > option_game > $o] : ( sup_su169929796game_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_128_sup_Ocommute,axiom,
( sup_su423136299n_game
= ( ^ [A4: set_Pr1625152599n_game,B4: set_Pr1625152599n_game] : ( sup_su423136299n_game @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_129_sup_Ocommute,axiom,
( sup_su1227468340game_o
= ( ^ [A4: set_variable > option_game > $o,B4: set_variable > option_game > $o] : ( sup_su1227468340game_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_130_sup_Ocommute,axiom,
( sup_su796825019n_game
= ( ^ [A4: set_Pr166476775n_game,B4: set_Pr166476775n_game] : ( sup_su796825019n_game @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_131_sup_Ocommute,axiom,
( sup_sup_set_variable
= ( ^ [A4: set_variable,B4: set_variable] : ( sup_sup_set_variable @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_132_sup__assoc,axiom,
! [X: variable > $o,Y: variable > $o,Z: variable > $o] :
( ( sup_sup_variable_o @ ( sup_sup_variable_o @ X @ Y ) @ Z )
= ( sup_sup_variable_o @ X @ ( sup_sup_variable_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_133_sup__assoc,axiom,
! [X: option_game > option_game > $o,Y: option_game > option_game > $o,Z: option_game > option_game > $o] :
( ( sup_su169929796game_o @ ( sup_su169929796game_o @ X @ Y ) @ Z )
= ( sup_su169929796game_o @ X @ ( sup_su169929796game_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_134_sup__assoc,axiom,
! [X: set_Pr1625152599n_game,Y: set_Pr1625152599n_game,Z: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ ( sup_su423136299n_game @ X @ Y ) @ Z )
= ( sup_su423136299n_game @ X @ ( sup_su423136299n_game @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_135_sup__assoc,axiom,
! [X: set_variable > option_game > $o,Y: set_variable > option_game > $o,Z: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ ( sup_su1227468340game_o @ X @ Y ) @ Z )
= ( sup_su1227468340game_o @ X @ ( sup_su1227468340game_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_136_sup__assoc,axiom,
! [X: set_Pr166476775n_game,Y: set_Pr166476775n_game,Z: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ ( sup_su796825019n_game @ X @ Y ) @ Z )
= ( sup_su796825019n_game @ X @ ( sup_su796825019n_game @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_137_sup__assoc,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_138_sup_Oassoc,axiom,
! [A2: variable > $o,B2: variable > $o,C2: variable > $o] :
( ( sup_sup_variable_o @ ( sup_sup_variable_o @ A2 @ B2 ) @ C2 )
= ( sup_sup_variable_o @ A2 @ ( sup_sup_variable_o @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_139_sup_Oassoc,axiom,
! [A2: option_game > option_game > $o,B2: option_game > option_game > $o,C2: option_game > option_game > $o] :
( ( sup_su169929796game_o @ ( sup_su169929796game_o @ A2 @ B2 ) @ C2 )
= ( sup_su169929796game_o @ A2 @ ( sup_su169929796game_o @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_140_sup_Oassoc,axiom,
! [A2: set_Pr1625152599n_game,B2: set_Pr1625152599n_game,C2: set_Pr1625152599n_game] :
( ( sup_su423136299n_game @ ( sup_su423136299n_game @ A2 @ B2 ) @ C2 )
= ( sup_su423136299n_game @ A2 @ ( sup_su423136299n_game @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_141_sup_Oassoc,axiom,
! [A2: set_variable > option_game > $o,B2: set_variable > option_game > $o,C2: set_variable > option_game > $o] :
( ( sup_su1227468340game_o @ ( sup_su1227468340game_o @ A2 @ B2 ) @ C2 )
= ( sup_su1227468340game_o @ A2 @ ( sup_su1227468340game_o @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_142_sup_Oassoc,axiom,
! [A2: set_Pr166476775n_game,B2: set_Pr166476775n_game,C2: set_Pr166476775n_game] :
( ( sup_su796825019n_game @ ( sup_su796825019n_game @ A2 @ B2 ) @ C2 )
= ( sup_su796825019n_game @ A2 @ ( sup_su796825019n_game @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_143_sup_Oassoc,axiom,
! [A2: set_variable,B2: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_144_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Pr166476775n_game,K: set_Pr166476775n_game,B2: set_Pr166476775n_game,A2: set_Pr166476775n_game] :
( ( B
= ( sup_su796825019n_game @ K @ B2 ) )
=> ( ( sup_su796825019n_game @ A2 @ B )
= ( sup_su796825019n_game @ K @ ( sup_su796825019n_game @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_145_boolean__algebra__cancel_Osup2,axiom,
! [B: set_variable,K: set_variable,B2: set_variable,A2: set_variable] :
( ( B
= ( sup_sup_set_variable @ K @ B2 ) )
=> ( ( sup_sup_set_variable @ A2 @ B )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_146_boolean__algebra__cancel_Osup1,axiom,
! [A: set_variable,K: set_variable,A2: set_variable,B2: set_variable] :
( ( A
= ( sup_sup_set_variable @ K @ A2 ) )
=> ( ( sup_sup_set_variable @ A @ B2 )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_147_inf__sup__aci_I5_J,axiom,
( sup_sup_set_variable
= ( ^ [X2: set_variable,Y3: set_variable] : ( sup_sup_set_variable @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_148_inf__sup__aci_I6_J,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_149_inf__sup__aci_I7_J,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) )
= ( sup_sup_set_variable @ Y @ ( sup_sup_set_variable @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_150_inf__sup__aci_I8_J,axiom,
! [X: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ X @ Y ) )
= ( sup_sup_set_variable @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_151_not__union__or,axiom,
! [X: variable,A: set_variable,B: set_variable] :
( ( ~ ( member_variable @ X @ ( sup_sup_set_variable @ A @ B ) ) )
= ( ~ ( member_variable @ X @ A )
& ~ ( member_variable @ X @ B ) ) ) ).
% not_union_or
thf(fact_152_Un__left__commute,axiom,
! [A: set_variable,B: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B @ C ) )
= ( sup_sup_set_variable @ B @ ( sup_sup_set_variable @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_153_Un__left__absorb,axiom,
! [A: set_variable,B: set_variable] :
( ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ A @ B ) )
= ( sup_sup_set_variable @ A @ B ) ) ).
% Un_left_absorb
thf(fact_154_Un__commute,axiom,
( sup_sup_set_variable
= ( ^ [A3: set_variable,B3: set_variable] : ( sup_sup_set_variable @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_155_Un__absorb,axiom,
! [A: set_variable] :
( ( sup_sup_set_variable @ A @ A )
= A ) ).
% Un_absorb
thf(fact_156_Un__assoc,axiom,
! [A: set_variable,B: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A @ B ) @ C )
= ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B @ C ) ) ) ).
% Un_assoc
thf(fact_157_ball__Un,axiom,
! [A: set_variable,B: set_variable,P: variable > $o] :
( ( ! [X2: variable] :
( ( member_variable @ X2 @ ( sup_sup_set_variable @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: variable] :
( ( member_variable @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: variable] :
( ( member_variable @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_158_bex__Un,axiom,
! [A: set_variable,B: set_variable,P: variable > $o] :
( ( ? [X2: variable] :
( ( member_variable @ X2 @ ( sup_sup_set_variable @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: variable] :
( ( member_variable @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: variable] :
( ( member_variable @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_159_UnI2,axiom,
! [C2: variable,B: set_variable,A: set_variable] :
( ( member_variable @ C2 @ B )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnI2
thf(fact_160_UnI1,axiom,
! [C2: variable,A: set_variable,B: set_variable] :
( ( member_variable @ C2 @ A )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnI1
thf(fact_161_UnE,axiom,
! [C2: variable,A: set_variable,B: set_variable] :
( ( member_variable @ C2 @ ( sup_sup_set_variable @ A @ B ) )
=> ( ~ ( member_variable @ C2 @ A )
=> ( member_variable @ C2 @ B ) ) ) ).
% UnE
thf(fact_162_usubstappp__antimon,axiom,
! [V: set_variable,U: set_variable,Sigma: produc1418842292n_game,Alpha: game] :
( ( ord_le282106107riable @ V @ U )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
=> ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
= ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_antimon
thf(fact_163_sup_OcoboundedI2,axiom,
! [C2: set_variable,B2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ C2 @ B2 )
=> ( ord_le282106107riable @ C2 @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_164_sup_OcoboundedI1,axiom,
! [C2: set_variable,A2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C2 @ A2 )
=> ( ord_le282106107riable @ C2 @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_165_sup_Oabsorb__iff2,axiom,
( ord_le282106107riable
= ( ^ [A4: set_variable,B4: set_variable] :
( ( sup_sup_set_variable @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_166_sup_Oabsorb__iff1,axiom,
( ord_le282106107riable
= ( ^ [B4: set_variable,A4: set_variable] :
( ( sup_sup_set_variable @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_167_sup_Ocobounded2,axiom,
! [B2: set_variable,A2: set_variable] : ( ord_le282106107riable @ B2 @ ( sup_sup_set_variable @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_168_sup_Ocobounded1,axiom,
! [A2: set_variable,B2: set_variable] : ( ord_le282106107riable @ A2 @ ( sup_sup_set_variable @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_169_sup_Oorder__iff,axiom,
( ord_le282106107riable
= ( ^ [B4: set_variable,A4: set_variable] :
( A4
= ( sup_sup_set_variable @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_170_sup_OboundedI,axiom,
! [B2: set_variable,A2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ B2 @ A2 )
=> ( ( ord_le282106107riable @ C2 @ A2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_171_sup_OboundedE,axiom,
! [B2: set_variable,C2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le282106107riable @ B2 @ A2 )
=> ~ ( ord_le282106107riable @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_172_sup__absorb2,axiom,
! [X: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X @ Y )
=> ( ( sup_sup_set_variable @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_173_sup__absorb1,axiom,
! [Y: set_variable,X: set_variable] :
( ( ord_le282106107riable @ Y @ X )
=> ( ( sup_sup_set_variable @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_174_sup_Oabsorb2,axiom,
! [A2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( sup_sup_set_variable @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_175_sup_Oabsorb1,axiom,
! [B2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B2 @ A2 )
=> ( ( sup_sup_set_variable @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_176_sup__unique,axiom,
! [F2: set_variable > set_variable > set_variable,X: set_variable,Y: set_variable] :
( ! [X3: set_variable,Y4: set_variable] : ( ord_le282106107riable @ X3 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: set_variable,Y4: set_variable] : ( ord_le282106107riable @ Y4 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: set_variable,Y4: set_variable,Z3: set_variable] :
( ( ord_le282106107riable @ Y4 @ X3 )
=> ( ( ord_le282106107riable @ Z3 @ X3 )
=> ( ord_le282106107riable @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_variable @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_177_sup_OorderI,axiom,
! [A2: set_variable,B2: set_variable] :
( ( A2
= ( sup_sup_set_variable @ A2 @ B2 ) )
=> ( ord_le282106107riable @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_178_sup_OorderE,axiom,
! [B2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_variable @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_179_le__iff__sup,axiom,
( ord_le282106107riable
= ( ^ [X2: set_variable,Y3: set_variable] :
( ( sup_sup_set_variable @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_180_sup__least,axiom,
! [Y: set_variable,X: set_variable,Z: set_variable] :
( ( ord_le282106107riable @ Y @ X )
=> ( ( ord_le282106107riable @ Z @ X )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_181_sup__mono,axiom,
! [A2: set_variable,C2: set_variable,B2: set_variable,D: set_variable] :
( ( ord_le282106107riable @ A2 @ C2 )
=> ( ( ord_le282106107riable @ B2 @ D )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A2 @ B2 ) @ ( sup_sup_set_variable @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_182_sup_Omono,axiom,
! [C2: set_variable,A2: set_variable,D: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C2 @ A2 )
=> ( ( ord_le282106107riable @ D @ B2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ C2 @ D ) @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_183_le__supI2,axiom,
! [X: set_variable,B2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ X @ B2 )
=> ( ord_le282106107riable @ X @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_184_le__supI1,axiom,
! [X: set_variable,A2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ X @ A2 )
=> ( ord_le282106107riable @ X @ ( sup_sup_set_variable @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_185_sup__ge2,axiom,
! [Y: set_variable,X: set_variable] : ( ord_le282106107riable @ Y @ ( sup_sup_set_variable @ X @ Y ) ) ).
% sup_ge2
thf(fact_186_sup__ge1,axiom,
! [X: set_variable,Y: set_variable] : ( ord_le282106107riable @ X @ ( sup_sup_set_variable @ X @ Y ) ) ).
% sup_ge1
thf(fact_187_le__supI,axiom,
! [A2: set_variable,X: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A2 @ X )
=> ( ( ord_le282106107riable @ B2 @ X )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_188_le__supE,axiom,
! [A2: set_variable,B2: set_variable,X: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le282106107riable @ A2 @ X )
=> ~ ( ord_le282106107riable @ B2 @ X ) ) ) ).
% le_supE
thf(fact_189_inf__sup__ord_I3_J,axiom,
! [X: set_variable,Y: set_variable] : ( ord_le282106107riable @ X @ ( sup_sup_set_variable @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_190_inf__sup__ord_I4_J,axiom,
! [Y: set_variable,X: set_variable] : ( ord_le282106107riable @ Y @ ( sup_sup_set_variable @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_191_subset__Un__eq,axiom,
( ord_le282106107riable
= ( ^ [A3: set_variable,B3: set_variable] :
( ( sup_sup_set_variable @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_192_subset__UnE,axiom,
! [C: set_variable,A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ C @ ( sup_sup_set_variable @ A @ B ) )
=> ~ ! [A5: set_variable] :
( ( ord_le282106107riable @ A5 @ A )
=> ! [B5: set_variable] :
( ( ord_le282106107riable @ B5 @ B )
=> ( C
!= ( sup_sup_set_variable @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_193_Un__absorb2,axiom,
! [B: set_variable,A: set_variable] :
( ( ord_le282106107riable @ B @ A )
=> ( ( sup_sup_set_variable @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_194_Un__absorb1,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( sup_sup_set_variable @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_195_Un__upper2,axiom,
! [B: set_variable,A: set_variable] : ( ord_le282106107riable @ B @ ( sup_sup_set_variable @ A @ B ) ) ).
% Un_upper2
thf(fact_196_Un__upper1,axiom,
! [A: set_variable,B: set_variable] : ( ord_le282106107riable @ A @ ( sup_sup_set_variable @ A @ B ) ) ).
% Un_upper1
thf(fact_197_Un__least,axiom,
! [A: set_variable,C: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ C )
=> ( ( ord_le282106107riable @ B @ C )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_198_Un__mono,axiom,
! [A: set_variable,C: set_variable,B: set_variable,D2: set_variable] :
( ( ord_le282106107riable @ A @ C )
=> ( ( ord_le282106107riable @ B @ D2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ ( sup_sup_set_variable @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_199_order__refl,axiom,
! [X: set_variable] : ( ord_le282106107riable @ X @ X ) ).
% order_refl
thf(fact_200_option_Oexpand,axiom,
! [Option: option_game,Option2: option_game] :
( ( ( Option = none_game )
= ( Option2 = none_game ) )
=> ( ( ( Option != none_game )
=> ( ( Option2 != none_game )
=> ( ( the_game @ Option )
= ( the_game @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_201_prod__eq__iff,axiom,
( ( ^ [Y2: produc1078154247n_game,Z2: produc1078154247n_game] : Y2 = Z2 )
= ( ^ [S: produc1078154247n_game,T: produc1078154247n_game] :
( ( ( produc893821739n_game @ S )
= ( produc893821739n_game @ T ) )
& ( ( produc293487213n_game @ S )
= ( produc293487213n_game @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_202_prod_Oexpand,axiom,
! [Prod: produc1078154247n_game,Prod2: produc1078154247n_game] :
( ( ( ( produc893821739n_game @ Prod )
= ( produc893821739n_game @ Prod2 ) )
& ( ( produc293487213n_game @ Prod )
= ( produc293487213n_game @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_203_exE__realizer_H,axiom,
! [P: option_game > set_variable > $o,P2: produc1078154247n_game] :
( ( P @ ( produc293487213n_game @ P2 ) @ ( produc893821739n_game @ P2 ) )
=> ~ ! [X3: set_variable,Y4: option_game] :
~ ( P @ Y4 @ X3 ) ) ).
% exE_realizer'
thf(fact_204_prod__eqI,axiom,
! [P2: produc1078154247n_game,Q2: produc1078154247n_game] :
( ( ( produc893821739n_game @ P2 )
= ( produc893821739n_game @ Q2 ) )
=> ( ( ( produc293487213n_game @ P2 )
= ( produc293487213n_game @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_205_usubstappp__loop__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
& ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
!= none_game ) ) ) ).
% usubstappp_loop_conv
thf(fact_206_BVG__choice,axiom,
! [Alpha: game,Beta: game] : ( ord_le282106107riable @ ( static_BVG @ ( choice @ Alpha @ Beta ) ) @ ( sup_sup_set_variable @ ( static_BVG @ Alpha ) @ ( static_BVG @ Beta ) ) ) ).
% BVG_choice
thf(fact_207_usubstappp__choice__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
& ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) )
!= none_game ) ) ) ).
% usubstappp_choice_conv
thf(fact_208_game_Oinject_I6_J,axiom,
! [X6: game,Y6: game] :
( ( ( loop @ X6 )
= ( loop @ Y6 ) )
= ( X6 = Y6 ) ) ).
% game.inject(6)
thf(fact_209_game_Oinject_I4_J,axiom,
! [X41: game,X42: game,Y41: game,Y42: game] :
( ( ( choice @ X41 @ X42 )
= ( choice @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% game.inject(4)
thf(fact_210_game_Odistinct_I39_J,axiom,
! [X41: game,X42: game,X6: game] :
( ( choice @ X41 @ X42 )
!= ( loop @ X6 ) ) ).
% game.distinct(39)
thf(fact_211_game_Odistinct_I45_J,axiom,
! [X51: game,X52: game,X6: game] :
( ( compose @ X51 @ X52 )
!= ( loop @ X6 ) ) ).
% game.distinct(45)
thf(fact_212_game_Odistinct_I37_J,axiom,
! [X41: game,X42: game,X51: game,X52: game] :
( ( choice @ X41 @ X42 )
!= ( compose @ X51 @ X52 ) ) ).
% game.distinct(37)
thf(fact_213_BVG__loop,axiom,
! [Alpha: game] : ( ord_le282106107riable @ ( static_BVG @ ( loop @ Alpha ) ) @ ( static_BVG @ Alpha ) ) ).
% BVG_loop
thf(fact_214_order__subst1,axiom,
! [A2: set_variable,F2: set_variable > set_variable,B2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C2 )
=> ( ! [X3: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X3 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_215_order__subst2,axiom,
! [A2: set_variable,B2: set_variable,F2: set_variable > set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( ord_le282106107riable @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X3 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_216_ord__eq__le__subst,axiom,
! [A2: set_variable,F2: set_variable > set_variable,B2: set_variable,C2: set_variable] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C2 )
=> ( ! [X3: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X3 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_217_ord__le__eq__subst,axiom,
! [A2: set_variable,B2: set_variable,F2: set_variable > set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X3 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_218_eq__iff,axiom,
( ( ^ [Y2: set_variable,Z2: set_variable] : Y2 = Z2 )
= ( ^ [X2: set_variable,Y3: set_variable] :
( ( ord_le282106107riable @ X2 @ Y3 )
& ( ord_le282106107riable @ Y3 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_219_antisym,axiom,
! [X: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X @ Y )
=> ( ( ord_le282106107riable @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_220_eq__refl,axiom,
! [X: set_variable,Y: set_variable] :
( ( X = Y )
=> ( ord_le282106107riable @ X @ Y ) ) ).
% eq_refl
thf(fact_221_order_Otrans,axiom,
! [A2: set_variable,B2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_222_antisym__conv,axiom,
! [Y: set_variable,X: set_variable] :
( ( ord_le282106107riable @ Y @ X )
=> ( ( ord_le282106107riable @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_223_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_variable,Z2: set_variable] : Y2 = Z2 )
= ( ^ [A4: set_variable,B4: set_variable] :
( ( ord_le282106107riable @ A4 @ B4 )
& ( ord_le282106107riable @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_224_ord__eq__le__trans,axiom,
! [A2: set_variable,B2: set_variable,C2: set_variable] :
( ( A2 = B2 )
=> ( ( ord_le282106107riable @ B2 @ C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_225_ord__le__eq__trans,axiom,
! [A2: set_variable,B2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_226_order__class_Oorder_Oantisym,axiom,
! [A2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_227_order__trans,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( ord_le282106107riable @ X @ Y )
=> ( ( ord_le282106107riable @ Y @ Z )
=> ( ord_le282106107riable @ X @ Z ) ) ) ).
% order_trans
thf(fact_228_dual__order_Orefl,axiom,
! [A2: set_variable] : ( ord_le282106107riable @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_229_dual__order_Otrans,axiom,
! [B2: set_variable,A2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ B2 @ A2 )
=> ( ( ord_le282106107riable @ C2 @ B2 )
=> ( ord_le282106107riable @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_230_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_variable,Z2: set_variable] : Y2 = Z2 )
= ( ^ [A4: set_variable,B4: set_variable] :
( ( ord_le282106107riable @ B4 @ A4 )
& ( ord_le282106107riable @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_231_dual__order_Oantisym,axiom,
! [B2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B2 @ A2 )
=> ( ( ord_le282106107riable @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_232_fst__swap,axiom,
! [X: produc735959047riable] :
( ( produc893821739n_game @ ( produc345397471riable @ X ) )
= ( produc284475501riable @ X ) ) ).
% fst_swap
thf(fact_233_fst__swap,axiom,
! [X: produc1078154247n_game] :
( ( produc884810027riable @ ( produc354409183n_game @ X ) )
= ( produc293487213n_game @ X ) ) ).
% fst_swap
thf(fact_234_snd__swap,axiom,
! [X: produc1078154247n_game] :
( ( produc284475501riable @ ( produc354409183n_game @ X ) )
= ( produc893821739n_game @ X ) ) ).
% snd_swap
thf(fact_235_snd__swap,axiom,
! [X: produc735959047riable] :
( ( produc293487213n_game @ ( produc345397471riable @ X ) )
= ( produc884810027riable @ X ) ) ).
% snd_swap
thf(fact_236_usubstappp__choice,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
= ( produc1149443391n_game @ ( sup_sup_set_variable @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1484167963hoiceo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% usubstappp_choice
thf(fact_237_GreatestI2__order,axiom,
! [P: set_variable > $o,X: set_variable,Q: set_variable > $o] :
( ( P @ X )
=> ( ! [Y4: set_variable] :
( ( P @ Y4 )
=> ( ord_le282106107riable @ Y4 @ X ) )
=> ( ! [X3: set_variable] :
( ( P @ X3 )
=> ( ! [Y5: set_variable] :
( ( P @ Y5 )
=> ( ord_le282106107riable @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_272405634riable @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_238_Greatest__equality,axiom,
! [P: set_variable > $o,X: set_variable] :
( ( P @ X )
=> ( ! [Y4: set_variable] :
( ( P @ Y4 )
=> ( ord_le282106107riable @ Y4 @ X ) )
=> ( ( order_272405634riable @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_239_old_Oprod_Oinject,axiom,
! [A2: set_variable,B2: option_game,A6: set_variable,B6: option_game] :
( ( ( produc1149443391n_game @ A2 @ B2 )
= ( produc1149443391n_game @ A6 @ B6 ) )
= ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_240_old_Oprod_Oinject,axiom,
! [A2: option_game,B2: option_game,A6: option_game,B6: option_game] :
( ( ( produc1111617711n_game @ A2 @ B2 )
= ( produc1111617711n_game @ A6 @ B6 ) )
= ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_241_prod_Oinject,axiom,
! [X1: set_variable,X22: option_game,Y1: set_variable,Y22: option_game] :
( ( ( produc1149443391n_game @ X1 @ X22 )
= ( produc1149443391n_game @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_242_prod_Oinject,axiom,
! [X1: option_game,X22: option_game,Y1: option_game,Y22: option_game] :
( ( ( produc1111617711n_game @ X1 @ X22 )
= ( produc1111617711n_game @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_243_swap__simp,axiom,
! [X: option_game,Y: set_variable] :
( ( produc345397471riable @ ( produc1140431679riable @ X @ Y ) )
= ( produc1149443391n_game @ Y @ X ) ) ).
% swap_simp
thf(fact_244_swap__simp,axiom,
! [X: set_variable,Y: option_game] :
( ( produc354409183n_game @ ( produc1149443391n_game @ X @ Y ) )
= ( produc1140431679riable @ Y @ X ) ) ).
% swap_simp
thf(fact_245_swap__simp,axiom,
! [X: option_game,Y: option_game] :
( ( produc1696194127n_game @ ( produc1111617711n_game @ X @ Y ) )
= ( produc1111617711n_game @ Y @ X ) ) ).
% swap_simp
thf(fact_246_prod_Ocollapse,axiom,
! [Prod: produc1016592119n_game] :
( ( produc1111617711n_game @ ( produc1942569115n_game @ Prod ) @ ( produc279011037n_game @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_247_prod_Ocollapse,axiom,
! [Prod: produc1078154247n_game] :
( ( produc1149443391n_game @ ( produc893821739n_game @ Prod ) @ ( produc293487213n_game @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_248_old_Oprod_Oinducts,axiom,
! [P: produc1078154247n_game > $o,Prod: produc1078154247n_game] :
( ! [A7: set_variable,B7: option_game] : ( P @ ( produc1149443391n_game @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_249_old_Oprod_Oinducts,axiom,
! [P: produc1016592119n_game > $o,Prod: produc1016592119n_game] :
( ! [A7: option_game,B7: option_game] : ( P @ ( produc1111617711n_game @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_250_old_Oprod_Oexhaust,axiom,
! [Y: produc1078154247n_game] :
~ ! [A7: set_variable,B7: option_game] :
( Y
!= ( produc1149443391n_game @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_251_old_Oprod_Oexhaust,axiom,
! [Y: produc1016592119n_game] :
~ ! [A7: option_game,B7: option_game] :
( Y
!= ( produc1111617711n_game @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_252_Pair__inject,axiom,
! [A2: set_variable,B2: option_game,A6: set_variable,B6: option_game] :
( ( ( produc1149443391n_game @ A2 @ B2 )
= ( produc1149443391n_game @ A6 @ B6 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_253_Pair__inject,axiom,
! [A2: option_game,B2: option_game,A6: option_game,B6: option_game] :
( ( ( produc1111617711n_game @ A2 @ B2 )
= ( produc1111617711n_game @ A6 @ B6 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_254_prod__cases,axiom,
! [P: produc1078154247n_game > $o,P2: produc1078154247n_game] :
( ! [A7: set_variable,B7: option_game] : ( P @ ( produc1149443391n_game @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_255_prod__cases,axiom,
! [P: produc1016592119n_game > $o,P2: produc1016592119n_game] :
( ! [A7: option_game,B7: option_game] : ( P @ ( produc1111617711n_game @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_256_surj__pair,axiom,
! [P2: produc1078154247n_game] :
? [X3: set_variable,Y4: option_game] :
( P2
= ( produc1149443391n_game @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_257_surj__pair,axiom,
! [P2: produc1016592119n_game] :
? [X3: option_game,Y4: option_game] :
( P2
= ( produc1111617711n_game @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_258_fst__eqD,axiom,
! [X: option_game,Y: option_game,A2: option_game] :
( ( ( produc1942569115n_game @ ( produc1111617711n_game @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_259_fst__eqD,axiom,
! [X: set_variable,Y: option_game,A2: set_variable] :
( ( ( produc893821739n_game @ ( produc1149443391n_game @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_260_fst__conv,axiom,
! [X1: option_game,X22: option_game] :
( ( produc1942569115n_game @ ( produc1111617711n_game @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_261_fst__conv,axiom,
! [X1: set_variable,X22: option_game] :
( ( produc893821739n_game @ ( produc1149443391n_game @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_262_snd__eqD,axiom,
! [X: option_game,Y: option_game,A2: option_game] :
( ( ( produc279011037n_game @ ( produc1111617711n_game @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_263_snd__eqD,axiom,
! [X: set_variable,Y: option_game,A2: option_game] :
( ( ( produc293487213n_game @ ( produc1149443391n_game @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_264_snd__conv,axiom,
! [X1: option_game,X22: option_game] :
( ( produc279011037n_game @ ( produc1111617711n_game @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_265_snd__conv,axiom,
! [X1: set_variable,X22: option_game] :
( ( produc293487213n_game @ ( produc1149443391n_game @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_266_prod_Oswap__def,axiom,
( produc345397471riable
= ( ^ [P3: produc735959047riable] : ( produc1149443391n_game @ ( produc284475501riable @ P3 ) @ ( produc884810027riable @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_267_prod_Oswap__def,axiom,
( produc1696194127n_game
= ( ^ [P3: produc1016592119n_game] : ( produc1111617711n_game @ ( produc279011037n_game @ P3 ) @ ( produc1942569115n_game @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_268_prod_Oswap__def,axiom,
( produc354409183n_game
= ( ^ [P3: produc1078154247n_game] : ( produc1140431679riable @ ( produc293487213n_game @ P3 ) @ ( produc893821739n_game @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_269_fst__pair,axiom,
! [A2: option_game,B2: option_game] :
( ( produc1942569115n_game @ ( produc1111617711n_game @ A2 @ B2 ) )
= A2 ) ).
% fst_pair
thf(fact_270_fst__pair,axiom,
! [A2: set_variable,B2: option_game] :
( ( produc893821739n_game @ ( produc1149443391n_game @ A2 @ B2 ) )
= A2 ) ).
% fst_pair
thf(fact_271_snd__pair,axiom,
! [A2: option_game,B2: option_game] :
( ( produc279011037n_game @ ( produc1111617711n_game @ A2 @ B2 ) )
= B2 ) ).
% snd_pair
thf(fact_272_snd__pair,axiom,
! [A2: set_variable,B2: option_game] :
( ( produc293487213n_game @ ( produc1149443391n_game @ A2 @ B2 ) )
= B2 ) ).
% snd_pair
thf(fact_273_surjective__pairing,axiom,
! [T2: produc1016592119n_game] :
( T2
= ( produc1111617711n_game @ ( produc1942569115n_game @ T2 ) @ ( produc279011037n_game @ T2 ) ) ) ).
% surjective_pairing
thf(fact_274_surjective__pairing,axiom,
! [T2: produc1078154247n_game] :
( T2
= ( produc1149443391n_game @ ( produc893821739n_game @ T2 ) @ ( produc293487213n_game @ T2 ) ) ) ).
% surjective_pairing
thf(fact_275_prod_Oexhaust__sel,axiom,
! [Prod: produc1016592119n_game] :
( Prod
= ( produc1111617711n_game @ ( produc1942569115n_game @ Prod ) @ ( produc279011037n_game @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_276_prod_Oexhaust__sel,axiom,
! [Prod: produc1078154247n_game] :
( Prod
= ( produc1149443391n_game @ ( produc893821739n_game @ Prod ) @ ( produc293487213n_game @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_277_conjI__realizer,axiom,
! [P: option_game > $o,P2: option_game,Q: option_game > $o,Q2: option_game] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc1942569115n_game @ ( produc1111617711n_game @ P2 @ Q2 ) ) )
& ( Q @ ( produc279011037n_game @ ( produc1111617711n_game @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_278_conjI__realizer,axiom,
! [P: set_variable > $o,P2: set_variable,Q: option_game > $o,Q2: option_game] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc893821739n_game @ ( produc1149443391n_game @ P2 @ Q2 ) ) )
& ( Q @ ( produc293487213n_game @ ( produc1149443391n_game @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_279_exI__realizer,axiom,
! [P: option_game > option_game > $o,Y: option_game,X: option_game] :
( ( P @ Y @ X )
=> ( P @ ( produc279011037n_game @ ( produc1111617711n_game @ X @ Y ) ) @ ( produc1942569115n_game @ ( produc1111617711n_game @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_280_exI__realizer,axiom,
! [P: option_game > set_variable > $o,Y: option_game,X: set_variable] :
( ( P @ Y @ X )
=> ( P @ ( produc293487213n_game @ ( produc1149443391n_game @ X @ Y ) ) @ ( produc893821739n_game @ ( produc1149443391n_game @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_281_Choiceo_Osimps_I2_J,axiom,
! [Alpha: option_game] :
( ( uSubst1484167963hoiceo @ Alpha @ none_game )
= none_game ) ).
% Choiceo.simps(2)
thf(fact_282_Choiceo__undef,axiom,
! [Alpha: option_game,Beta: option_game] :
( ( ( uSubst1484167963hoiceo @ Alpha @ Beta )
= none_game )
= ( ( Alpha = none_game )
| ( Beta = none_game ) ) ) ).
% Choiceo_undef
thf(fact_283_usubstappp__compose,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( compose @ Alpha @ Beta ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) @ ( uSubst532817840mposeo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) ) ) ) ).
% usubstappp_compose
thf(fact_284_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: option_game > option_game > $o,X: option_game,Y: option_game,A2: produc1016592119n_game] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1111617711n_game @ X @ Y ) )
=> ( P @ ( produc1942569115n_game @ A2 ) @ ( produc279011037n_game @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_285_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: set_variable > option_game > $o,X: set_variable,Y: option_game,A2: produc1078154247n_game] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1149443391n_game @ X @ Y ) )
=> ( P @ ( produc893821739n_game @ A2 ) @ ( produc293487213n_game @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_286_Composeo__undef,axiom,
! [Alpha: option_game,Beta: option_game] :
( ( ( uSubst532817840mposeo @ Alpha @ Beta )
= none_game )
= ( ( Alpha = none_game )
| ( Beta = none_game ) ) ) ).
% Composeo_undef
thf(fact_287_Composeo_Osimps_I2_J,axiom,
! [Alpha: option_game] :
( ( uSubst532817840mposeo @ Alpha @ none_game )
= none_game ) ).
% Composeo.simps(2)
thf(fact_288_usubstappp__loop,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst23177304_Loopo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp_loop
thf(fact_289_eq__snd__iff,axiom,
! [B2: option_game,P2: produc1016592119n_game] :
( ( B2
= ( produc279011037n_game @ P2 ) )
= ( ? [A4: option_game] :
( P2
= ( produc1111617711n_game @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_290_eq__snd__iff,axiom,
! [B2: option_game,P2: produc1078154247n_game] :
( ( B2
= ( produc293487213n_game @ P2 ) )
= ( ? [A4: set_variable] :
( P2
= ( produc1149443391n_game @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_291_sndI,axiom,
! [X: produc1016592119n_game,Y: option_game,Z: option_game] :
( ( X
= ( produc1111617711n_game @ Y @ Z ) )
=> ( ( produc279011037n_game @ X )
= Z ) ) ).
% sndI
thf(fact_292_sndI,axiom,
! [X: produc1078154247n_game,Y: set_variable,Z: option_game] :
( ( X
= ( produc1149443391n_game @ Y @ Z ) )
=> ( ( produc293487213n_game @ X )
= Z ) ) ).
% sndI
thf(fact_293_Loopo_Osimps_I2_J,axiom,
( ( uSubst23177304_Loopo @ none_game )
= none_game ) ).
% Loopo.simps(2)
thf(fact_294_Loopo__undef,axiom,
! [Alpha: option_game] :
( ( ( uSubst23177304_Loopo @ Alpha )
= none_game )
= ( Alpha = none_game ) ) ).
% Loopo_undef
thf(fact_295_fstI,axiom,
! [X: produc1016592119n_game,Y: option_game,Z: option_game] :
( ( X
= ( produc1111617711n_game @ Y @ Z ) )
=> ( ( produc1942569115n_game @ X )
= Y ) ) ).
% fstI
thf(fact_296_fstI,axiom,
! [X: produc1078154247n_game,Y: set_variable,Z: option_game] :
( ( X
= ( produc1149443391n_game @ Y @ Z ) )
=> ( ( produc893821739n_game @ X )
= Y ) ) ).
% fstI
thf(fact_297_eq__fst__iff,axiom,
! [A2: option_game,P2: produc1016592119n_game] :
( ( A2
= ( produc1942569115n_game @ P2 ) )
= ( ? [B4: option_game] :
( P2
= ( produc1111617711n_game @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_298_eq__fst__iff,axiom,
! [A2: set_variable,P2: produc1078154247n_game] :
( ( A2
= ( produc893821739n_game @ P2 ) )
= ( ? [B4: option_game] :
( P2
= ( produc1149443391n_game @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_299_relChain__def,axiom,
( bNF_Ca1986151756riable
= ( ^ [R: set_Pr1625152599n_game,As: option_game > set_variable] :
! [I: option_game,J: option_game] :
( ( member191809696n_game @ ( produc1111617711n_game @ I @ J ) @ R )
=> ( ord_le282106107riable @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_300_sndOp__def,axiom,
( bNF_sn582998327n_game
= ( ^ [P4: set_variable > set_variable > $o,Q3: set_variable > option_game > $o,Ac: produc1078154247n_game] : ( produc1149443391n_game @ ( bNF_pi1813424679n_game @ P4 @ Q3 @ ( produc893821739n_game @ Ac ) @ ( produc293487213n_game @ Ac ) ) @ ( produc293487213n_game @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_301_sndOp__def,axiom,
( bNF_sn1374695591n_game
= ( ^ [P4: set_variable > option_game > $o,Q3: option_game > option_game > $o,Ac: produc1078154247n_game] : ( produc1111617711n_game @ ( bNF_pi856525207n_game @ P4 @ Q3 @ ( produc893821739n_game @ Ac ) @ ( produc293487213n_game @ Ac ) ) @ ( produc293487213n_game @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_302_fstOp__def,axiom,
( bNF_fs984229733n_game
= ( ^ [P4: set_variable > option_game > $o,Q3: option_game > option_game > $o,Ac: produc1078154247n_game] : ( produc1149443391n_game @ ( produc893821739n_game @ Ac ) @ ( bNF_pi856525207n_game @ P4 @ Q3 @ ( produc893821739n_game @ Ac ) @ ( produc293487213n_game @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_303_usubstappp_Osimps_I4_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
= ( produc1149443391n_game @ ( sup_sup_set_variable @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1484167963hoiceo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% usubstappp.simps(4)
thf(fact_304_less__eq__set__def,axiom,
( ord_le282106107riable
= ( ^ [A3: set_variable,B3: set_variable] :
( ord_le1407353162able_o
@ ^ [X2: variable] : ( member_variable @ X2 @ A3 )
@ ^ [X2: variable] : ( member_variable @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_305_Collect__subset,axiom,
! [A: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_306_Un__def,axiom,
( sup_sup_set_variable
= ( ^ [A3: set_variable,B3: set_variable] :
( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ A3 )
| ( member_variable @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_307_sup__set__def,axiom,
( sup_sup_set_variable
= ( ^ [A3: set_variable,B3: set_variable] :
( collect_variable
@ ( sup_sup_variable_o
@ ^ [X2: variable] : ( member_variable @ X2 @ A3 )
@ ^ [X2: variable] : ( member_variable @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_308_Collect__disj__eq,axiom,
! [P: variable > $o,Q: variable > $o] :
( ( collect_variable
@ ^ [X2: variable] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_variable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_309_subset__CollectI,axiom,
! [B: set_variable,A: set_variable,Q: variable > $o,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ! [X3: variable] :
( ( member_variable @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le282106107riable
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_310_subset__Collect__iff,axiom,
! [B: set_variable,A: set_variable,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ( ord_le282106107riable @ B
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: variable] :
( ( member_variable @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_311_usubstappp_Osimps_I5_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( compose @ Alpha @ Beta ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) @ ( uSubst532817840mposeo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) ) ) ) ).
% usubstappp.simps(5)
thf(fact_312_usubstappp_Osimps_I6_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst23177304_Loopo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp.simps(6)
thf(fact_313_sup__Un__eq,axiom,
! [R2: set_variable,S2: set_variable] :
( ( sup_sup_variable_o
@ ^ [X2: variable] : ( member_variable @ X2 @ R2 )
@ ^ [X2: variable] : ( member_variable @ X2 @ S2 ) )
= ( ^ [X2: variable] : ( member_variable @ X2 @ ( sup_sup_set_variable @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_314_subrelI,axiom,
! [R3: set_Pr166476775n_game,S3: set_Pr166476775n_game] :
( ! [X3: set_variable,Y4: option_game] :
( ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y4 ) @ R3 )
=> ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y4 ) @ S3 ) )
=> ( ord_le17855367n_game @ R3 @ S3 ) ) ).
% subrelI
thf(fact_315_subrelI,axiom,
! [R3: set_Pr1625152599n_game,S3: set_Pr1625152599n_game] :
( ! [X3: option_game,Y4: option_game] :
( ( member191809696n_game @ ( produc1111617711n_game @ X3 @ Y4 ) @ R3 )
=> ( member191809696n_game @ ( produc1111617711n_game @ X3 @ Y4 ) @ S3 ) )
=> ( ord_le1780499447n_game @ R3 @ S3 ) ) ).
% subrelI
thf(fact_316_pred__equals__eq2,axiom,
! [R2: set_Pr166476775n_game,S2: set_Pr166476775n_game] :
( ( ( ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_317_pred__equals__eq2,axiom,
! [R2: set_Pr1625152599n_game,S2: set_Pr1625152599n_game] :
( ( ( ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_318_pred__subset__eq2,axiom,
! [R2: set_Pr166476775n_game,S2: set_Pr166476775n_game] :
( ( ord_le2134856704game_o
@ ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ R2 )
@ ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ S2 ) )
= ( ord_le17855367n_game @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_319_pred__subset__eq2,axiom,
! [R2: set_Pr1625152599n_game,S2: set_Pr1625152599n_game] :
( ( ord_le707292176game_o
@ ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ R2 )
@ ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ S2 ) )
= ( ord_le1780499447n_game @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_320_sup__Un__eq2,axiom,
! [R2: set_Pr166476775n_game,S2: set_Pr166476775n_game] :
( ( sup_su1227468340game_o
@ ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ R2 )
@ ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ S2 ) )
= ( ^ [X2: set_variable,Y3: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X2 @ Y3 ) @ ( sup_su796825019n_game @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_321_sup__Un__eq2,axiom,
! [R2: set_Pr1625152599n_game,S2: set_Pr1625152599n_game] :
( ( sup_su169929796game_o
@ ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ R2 )
@ ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ S2 ) )
= ( ^ [X2: option_game,Y3: option_game] : ( member191809696n_game @ ( produc1111617711n_game @ X2 @ Y3 ) @ ( sup_su423136299n_game @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_322_pred__subset__eq,axiom,
! [R2: set_variable,S2: set_variable] :
( ( ord_le1407353162able_o
@ ^ [X2: variable] : ( member_variable @ X2 @ R2 )
@ ^ [X2: variable] : ( member_variable @ X2 @ S2 ) )
= ( ord_le282106107riable @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_323_conj__subset__def,axiom,
! [A: set_variable,P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ A
@ ( collect_variable
@ ^ [X2: variable] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le282106107riable @ A @ ( collect_variable @ P ) )
& ( ord_le282106107riable @ A @ ( collect_variable @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_324_prop__restrict,axiom,
! [X: variable,Z4: set_variable,X4: set_variable,P: variable > $o] :
( ( member_variable @ X @ Z4 )
=> ( ( ord_le282106107riable @ Z4
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_325_ssubst__Pair__rhs,axiom,
! [R3: set_variable,S3: option_game,R2: set_Pr166476775n_game,S4: option_game] :
( ( member171223600n_game @ ( produc1149443391n_game @ R3 @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member171223600n_game @ ( produc1149443391n_game @ R3 @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_326_ssubst__Pair__rhs,axiom,
! [R3: option_game,S3: option_game,R2: set_Pr1625152599n_game,S4: option_game] :
( ( member191809696n_game @ ( produc1111617711n_game @ R3 @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member191809696n_game @ ( produc1111617711n_game @ R3 @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_327_Collect__restrict,axiom,
! [X4: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X2: variable] :
( ( member_variable @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_328_Loopo_Oelims,axiom,
! [X: option_game,Y: option_game] :
( ( ( uSubst23177304_Loopo @ X )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some_game @ Alpha2 ) )
=> ( Y
!= ( some_game @ ( loop @ Alpha2 ) ) ) )
=> ~ ( ( X = none_game )
=> ( Y != none_game ) ) ) ) ).
% Loopo.elims
thf(fact_329_option_Oinject,axiom,
! [X22: game,Y22: game] :
( ( ( some_game @ X22 )
= ( some_game @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_330_not__Some__eq,axiom,
! [X: option_game] :
( ( ! [Y3: game] :
( X
!= ( some_game @ Y3 ) ) )
= ( X = none_game ) ) ).
% not_Some_eq
thf(fact_331_not__None__eq,axiom,
! [X: option_game] :
( ( X != none_game )
= ( ? [Y3: game] :
( X
= ( some_game @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_332_option_Ocollapse,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( ( some_game @ ( the_game @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_333_Composeo_Ocases,axiom,
! [X: produc1016592119n_game] :
( ! [Alpha2: game,Beta2: game] :
( X
!= ( produc1111617711n_game @ ( some_game @ Alpha2 ) @ ( some_game @ Beta2 ) ) )
=> ( ! [Alpha2: option_game] :
( X
!= ( produc1111617711n_game @ Alpha2 @ none_game ) )
=> ~ ! [V2: game] :
( X
!= ( produc1111617711n_game @ none_game @ ( some_game @ V2 ) ) ) ) ) ).
% Composeo.cases
thf(fact_334_option_Oexhaust__sel,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( Option
= ( some_game @ ( the_game @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_335_combine__options__cases,axiom,
! [X: option_game,P: option_game > option_game > $o,Y: option_game] :
( ( ( X = none_game )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_game )
=> ( P @ X @ Y ) )
=> ( ! [A7: game,B7: game] :
( ( X
= ( some_game @ A7 ) )
=> ( ( Y
= ( some_game @ B7 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_336_split__option__all,axiom,
( ( ^ [P5: option_game > $o] :
! [X5: option_game] : ( P5 @ X5 ) )
= ( ^ [P4: option_game > $o] :
( ( P4 @ none_game )
& ! [X2: game] : ( P4 @ ( some_game @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_337_split__option__ex,axiom,
( ( ^ [P5: option_game > $o] :
? [X5: option_game] : ( P5 @ X5 ) )
= ( ^ [P4: option_game > $o] :
( ( P4 @ none_game )
| ? [X2: game] : ( P4 @ ( some_game @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_338_option_Oinducts,axiom,
! [P: option_game > $o,Option: option_game] :
( ( P @ none_game )
=> ( ! [X3: game] : ( P @ ( some_game @ X3 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_339_option_Oexhaust,axiom,
! [Y: option_game] :
( ( Y != none_game )
=> ~ ! [X23: game] :
( Y
!= ( some_game @ X23 ) ) ) ).
% option.exhaust
thf(fact_340_option_OdiscI,axiom,
! [Option: option_game,X22: game] :
( ( Option
= ( some_game @ X22 ) )
=> ( Option != none_game ) ) ).
% option.discI
thf(fact_341_option_Odistinct_I1_J,axiom,
! [X22: game] :
( none_game
!= ( some_game @ X22 ) ) ).
% option.distinct(1)
thf(fact_342_option_Osel,axiom,
! [X22: game] :
( ( the_game @ ( some_game @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_343_Composeo_Oinduct,axiom,
! [P: option_game > option_game > $o,A0: option_game,A1: option_game] :
( ! [Alpha2: game,Beta2: game] : ( P @ ( some_game @ Alpha2 ) @ ( some_game @ Beta2 ) )
=> ( ! [Alpha2: option_game] : ( P @ Alpha2 @ none_game )
=> ( ! [V2: game] : ( P @ none_game @ ( some_game @ V2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_344_undefg__equiv,axiom,
! [Alpha: option_game] :
( ( Alpha != none_game )
= ( ? [G: game] :
( Alpha
= ( some_game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_345_Loopo_Oinduct,axiom,
! [P: option_game > $o,A0: option_game] :
( ! [Alpha2: game] : ( P @ ( some_game @ Alpha2 ) )
=> ( ( P @ none_game )
=> ( P @ A0 ) ) ) ).
% Loopo.induct
thf(fact_346_Loopo_Ocases,axiom,
! [X: option_game] :
( ! [Alpha2: game] :
( X
!= ( some_game @ Alpha2 ) )
=> ( X = none_game ) ) ).
% Loopo.cases
thf(fact_347_Loopo_Osimps_I1_J,axiom,
! [Alpha: game] :
( ( uSubst23177304_Loopo @ ( some_game @ Alpha ) )
= ( some_game @ ( loop @ Alpha ) ) ) ).
% Loopo.simps(1)
thf(fact_348_Choiceo_Osimps_I1_J,axiom,
! [Alpha: game,Beta: game] :
( ( uSubst1484167963hoiceo @ ( some_game @ Alpha ) @ ( some_game @ Beta ) )
= ( some_game @ ( choice @ Alpha @ Beta ) ) ) ).
% Choiceo.simps(1)
thf(fact_349_Composeo_Osimps_I1_J,axiom,
! [Alpha: game,Beta: game] :
( ( uSubst532817840mposeo @ ( some_game @ Alpha ) @ ( some_game @ Beta ) )
= ( some_game @ ( compose @ Alpha @ Beta ) ) ) ).
% Composeo.simps(1)
thf(fact_350_Composeo_Osimps_I3_J,axiom,
! [V3: game] :
( ( uSubst532817840mposeo @ none_game @ ( some_game @ V3 ) )
= none_game ) ).
% Composeo.simps(3)
thf(fact_351_Choiceo_Osimps_I3_J,axiom,
! [V3: game] :
( ( uSubst1484167963hoiceo @ none_game @ ( some_game @ V3 ) )
= none_game ) ).
% Choiceo.simps(3)
thf(fact_352_Composeo_Oelims,axiom,
! [X: option_game,Xa: option_game,Y: option_game] :
( ( ( uSubst532817840mposeo @ X @ Xa )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some_game @ Alpha2 ) )
=> ! [Beta2: game] :
( ( Xa
= ( some_game @ Beta2 ) )
=> ( Y
!= ( some_game @ ( compose @ Alpha2 @ Beta2 ) ) ) ) )
=> ( ( ( Xa = none_game )
=> ( Y != none_game ) )
=> ~ ( ( X = none_game )
=> ( ? [V2: game] :
( Xa
= ( some_game @ V2 ) )
=> ( Y != none_game ) ) ) ) ) ) ).
% Composeo.elims
thf(fact_353_Choiceo_Oelims,axiom,
! [X: option_game,Xa: option_game,Y: option_game] :
( ( ( uSubst1484167963hoiceo @ X @ Xa )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some_game @ Alpha2 ) )
=> ! [Beta2: game] :
( ( Xa
= ( some_game @ Beta2 ) )
=> ( Y
!= ( some_game @ ( choice @ Alpha2 @ Beta2 ) ) ) ) )
=> ( ( ( Xa = none_game )
=> ( Y != none_game ) )
=> ~ ( ( X = none_game )
=> ( ? [V2: game] :
( Xa
= ( some_game @ V2 ) )
=> ( Y != none_game ) ) ) ) ) ) ).
% Choiceo.elims
% Conjectures (1)
thf(conj_0,conjecture,
ord_le282106107riable @ ( sup_sup_set_variable @ ua @ ( static_BVG @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) ) ) ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
%------------------------------------------------------------------------------