TPTP Problem File: SLH0591^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00420_015537__12241458_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1445 ( 585 unt; 165 typ; 0 def)
% Number of atoms : 3750 (1327 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11308 ( 445 ~; 62 |; 308 &;8728 @)
% ( 0 <=>;1765 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 702 ( 702 >; 0 *; 0 +; 0 <<)
% Number of symbols : 155 ( 152 usr; 24 con; 0-5 aty)
% Number of variables : 3415 ( 151 ^;3149 !; 115 ?;3415 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:21:21.042
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (152)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Real__Oreal_J,type,
finite9007344921179782393t_real: set_set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
inf_inf_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
lattic8718645017227715691in_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
lattic3629708407755379051n_real: set_real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_ORuzsa__distance_001tf__a,type,
pluenn5761198478017115492ance_a: set_a > ( a > a > a ) > a > set_a > set_a > real ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001tf__a,type,
pluenn2534204936789923946sset_a: set_a > ( a > a > a ) > a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__Int__Oint,type,
pow_int: set_int > set_set_int ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Real__Oreal,type,
pow_real: set_real > set_set_real ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__a_J,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Real__Oreal,type,
image_int_real: ( int > real ) > set_int > set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001tf__a,type,
image_int_a: ( int > a ) > set_int > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_Itf__a_J,type,
image_real_set_a: ( real > set_a ) > set_real > set_set_a ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001tf__a,type,
image_real_a: ( real > a ) > set_real > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_3546087905283185883t_real: ( set_set_a > set_real ) > set_set_set_a > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
image_set_a_real: ( set_a > real ) > set_set_a > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Real__Oreal,type,
image_a_real: ( a > real ) > set_a > set_real ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
insert_set_int: set_int > set_set_int > set_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Real__Oreal_J,type,
insert_set_real: set_real > set_set_real > set_set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
insert_set_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_fChoice_001t__Set__Oset_Itf__a_J,type,
fChoice_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A0,type,
a0: set_a ).
thf(sy_v_A____,type,
a2: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_K0,type,
k0: real ).
thf(sy_v_KS____,type,
ks: set_real ).
thf(sy_v_K____,type,
k: real ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_thesisa____,type,
thesisa: $o ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1274)
thf(fact_0_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_1_that,axiom,
( ( finite_finite_real @ ks )
=> ( ( ks != bot_bot_set_real )
=> thesisa ) ) ).
% that
thf(fact_2_K__def,axiom,
( k
= ( lattic3629708407755379051n_real @ ks ) ) ).
% K_def
thf(fact_3_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_4_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_5_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_6_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_7_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_8_all__not__in__conv,axiom,
! [A: set_int] :
( ( ! [X: int] :
~ ( member_int @ X @ A ) )
= ( A = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_9_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X: real] :
~ ( member_real @ X @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_10_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_12_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_13_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X: int] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_14_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X: real] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_15_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_16_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_17_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X: int] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X: real] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_22_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_23_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_24_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_25_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_26_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_27_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_28_infinite__imp__nonempty,axiom,
! [S: set_real] :
( ~ ( finite_finite_real @ S )
=> ( S != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_29_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_30_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_31_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_32_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_33_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_34_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_35_finite__has__maximal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_36_finite__has__maximal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_37_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_38_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_39_finite__has__minimal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_40_finite__has__minimal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_41_finite_Ocases,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ~ ! [A3: set_real] :
( ? [A4: real] :
( A2
= ( insert_real @ A4 @ A3 ) )
=> ~ ( finite_finite_real @ A3 ) ) ) ) ).
% finite.cases
thf(fact_42_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A3: set_a] :
( ? [A4: a] :
( A2
= ( insert_a @ A4 @ A3 ) )
=> ~ ( finite_finite_a @ A3 ) ) ) ) ).
% finite.cases
thf(fact_43_finite_Ocases,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ~ ! [A3: set_set_a] :
( ? [A4: set_a] :
( A2
= ( insert_set_a @ A4 @ A3 ) )
=> ~ ( finite_finite_set_a @ A3 ) ) ) ) ).
% finite.cases
thf(fact_44_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A3: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A3 ) )
=> ~ ( finite_finite_nat @ A3 ) ) ) ) ).
% finite.cases
thf(fact_45_finite_Ocases,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ~ ! [A3: set_int] :
( ? [A4: int] :
( A2
= ( insert_int @ A4 @ A3 ) )
=> ~ ( finite_finite_int @ A3 ) ) ) ) ).
% finite.cases
thf(fact_46_assms_I3_J,axiom,
ord_less_eq_set_a @ a0 @ g ).
% assms(3)
thf(fact_47_assms_I7_J,axiom,
b != bot_bot_set_a ).
% assms(7)
thf(fact_48_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_49_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_50_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_51_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_52_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_53_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_54_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_55_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_56_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_57_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_58_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_59_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_60_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_61_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_62_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_63_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_64_insertCI,axiom,
! [A2: real,B: set_real,B2: real] :
( ( ~ ( member_real @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_65_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_66_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_67_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_68_insert__iff,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_69_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_70_insert__absorb2,axiom,
! [X3: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A ) )
= ( insert_a @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_71_insert__absorb2,axiom,
! [X3: set_a,A: set_set_a] :
( ( insert_set_a @ X3 @ ( insert_set_a @ X3 @ A ) )
= ( insert_set_a @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_72_commutative,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X3 @ Y )
= ( addition @ Y @ X3 ) ) ) ) ).
% commutative
thf(fact_73_empty__subsetI,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% empty_subsetI
thf(fact_74_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_75_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_76_empty__subsetI,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% empty_subsetI
thf(fact_77_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_78_subset__empty,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_79_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_80_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_81_subset__empty,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_82_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_83_singletonI,axiom,
! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_84_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_85_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_86_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_87_singletonI,axiom,
! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_88_finite__insert,axiom,
! [A2: real,A: set_real] :
( ( finite_finite_real @ ( insert_real @ A2 @ A ) )
= ( finite_finite_real @ A ) ) ).
% finite_insert
thf(fact_89_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_90_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_91_finite__insert,axiom,
! [A2: int,A: set_int] :
( ( finite_finite_int @ ( insert_int @ A2 @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_insert
thf(fact_92_finite__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) )
= ( finite_finite_set_a @ A ) ) ).
% finite_insert
thf(fact_93_insert__subset,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( ( member_set_a @ X3 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_94_insert__subset,axiom,
! [X3: real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A ) @ B )
= ( ( member_real @ X3 @ B )
& ( ord_less_eq_set_real @ A @ B ) ) ) ).
% insert_subset
thf(fact_95_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_96_insert__subset,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_97_singleton__insert__inj__eq_H,axiom,
! [A2: real,A: set_real,B2: real] :
( ( ( insert_real @ A2 @ A )
= ( insert_real @ B2 @ bot_bot_set_real ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_98_singleton__insert__inj__eq_H,axiom,
! [A2: set_a,A: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_99_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_100_singleton__insert__inj__eq_H,axiom,
! [A2: int,A: set_int,B2: int] :
( ( ( insert_int @ A2 @ A )
= ( insert_int @ B2 @ bot_bot_set_int ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_101_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_102_singleton__insert__inj__eq,axiom,
! [B2: real,A2: real,A: set_real] :
( ( ( insert_real @ B2 @ bot_bot_set_real )
= ( insert_real @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_103_singleton__insert__inj__eq,axiom,
! [B2: set_a,A2: set_a,A: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_104_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_105_singleton__insert__inj__eq,axiom,
! [B2: int,A2: int,A: set_int] :
( ( ( insert_int @ B2 @ bot_bot_set_int )
= ( insert_int @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_106_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_107_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_108_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_109_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_110_le__cases3,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_111_le__cases3,axiom,
! [X3: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_112_le__cases3,axiom,
! [X3: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_113_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_114_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_115_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_116_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
& ( ord_less_eq_int @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_117_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_118_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_119_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_120_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_121_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_122_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_123_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_124_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_125_order__antisym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_126_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_127_order__antisym,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_128_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_129_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_130_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_131_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_132_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_133_order__trans,axiom,
! [X3: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_135_order__trans,axiom,
! [X3: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_136_order__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_137_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_138_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_139_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_140_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_141_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_142_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_143_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_144_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_145_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_146_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_149_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_150_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_151_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_152_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_153_in__mono,axiom,
! [A: set_real,B: set_real,X3: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X3 @ A )
=> ( member_real @ X3 @ B ) ) ) ).
% in_mono
thf(fact_154_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_155_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_156_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_166_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_167_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_168_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_169_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_170_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_171_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_172_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_173_insertE,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_174_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_175_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_176_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_177_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_178_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_179_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_180_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_181_insertI1,axiom,
! [A2: real,B: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B ) ) ).
% insertI1
thf(fact_182_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_183_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_184_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_185_insertI2,axiom,
! [A2: real,B: set_real,B2: real] :
( ( member_real @ A2 @ B )
=> ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_186_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_187_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_188_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B5: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A6 )
=> ( member_set_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_189_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B5: set_real] :
! [X: real] :
( ( member_real @ X @ A6 )
=> ( member_real @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_190_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A6 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_191_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X: a] :
( ( member_a @ X @ A6 )
=> ( member_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_192_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_193_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_194_Set_Oset__insert,axiom,
! [X3: a,A: set_a] :
( ( member_a @ X3 @ A )
=> ~ ! [B6: set_a] :
( ( A
= ( insert_a @ X3 @ B6 ) )
=> ( member_a @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_195_Set_Oset__insert,axiom,
! [X3: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ! [B6: set_set_a] :
( ( A
= ( insert_set_a @ X3 @ B6 ) )
=> ( member_set_a @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_196_Set_Oset__insert,axiom,
! [X3: real,A: set_real] :
( ( member_real @ X3 @ A )
=> ~ ! [B6: set_real] :
( ( A
= ( insert_real @ X3 @ B6 ) )
=> ( member_real @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_197_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat @ X3 @ B6 ) )
=> ( member_nat @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_198_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A6 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_199_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B5: set_real] :
! [T: real] :
( ( member_real @ T @ A6 )
=> ( member_real @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_200_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A6 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_201_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_202_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C2 ) @ ( insert_set_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_203_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_204_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_205_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_206_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_207_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_208_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_209_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_210_insert__ident,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B )
=> ( ( ( insert_a @ X3 @ A )
= ( insert_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_211_insert__ident,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ~ ( member_set_a @ X3 @ B )
=> ( ( ( insert_set_a @ X3 @ A )
= ( insert_set_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_212_insert__ident,axiom,
! [X3: real,A: set_real,B: set_real] :
( ~ ( member_real @ X3 @ A )
=> ( ~ ( member_real @ X3 @ B )
=> ( ( ( insert_real @ X3 @ A )
= ( insert_real @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_213_insert__ident,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_214_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_215_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_216_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_217_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_218_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_219_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_220_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_221_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_222_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_223_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_224_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_225_insert__eq__iff,axiom,
! [A2: real,A: set_real,B2: real,B: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ~ ( member_real @ B2 @ B )
=> ( ( ( insert_real @ A2 @ A )
= ( insert_real @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_real] :
( ( A
= ( insert_real @ B2 @ C3 ) )
& ~ ( member_real @ B2 @ C3 )
& ( B
= ( insert_real @ A2 @ C3 ) )
& ~ ( member_real @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_226_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_227_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_228_subset__insert,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_229_subset__insert,axiom,
! [X3: real,A: set_real,B: set_real] :
( ~ ( member_real @ X3 @ A )
=> ( ( ord_less_eq_set_real @ A @ ( insert_real @ X3 @ B ) )
= ( ord_less_eq_set_real @ A @ B ) ) ) ).
% subset_insert
thf(fact_230_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_231_subset__insert,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_232_insert__commute,axiom,
! [X3: a,Y: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ Y @ A ) )
= ( insert_a @ Y @ ( insert_a @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_233_insert__commute,axiom,
! [X3: set_a,Y: set_a,A: set_set_a] :
( ( insert_set_a @ X3 @ ( insert_set_a @ Y @ A ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_234_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_235_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_236_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_237_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_238_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_239_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_240_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_241_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X: real] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_242_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_243_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_244_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_245_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_246_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_247_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_248_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_249_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_250_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_251_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_252_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_253_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_254_order__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_255_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_256_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_257_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_258_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_259_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_260_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_261_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_262_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_263_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_264_order__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_265_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_266_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_267_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B6: set_a] :
( ( A
= ( insert_a @ A2 @ B6 ) )
& ~ ( member_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_268_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B6: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B6 ) )
& ~ ( member_set_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_269_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ? [B6: set_real] :
( ( A
= ( insert_real @ A2 @ B6 ) )
& ~ ( member_real @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_270_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat @ A2 @ B6 ) )
& ~ ( member_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_271_order__eq__refl,axiom,
! [X3: set_a,Y: set_a] :
( ( X3 = Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_272_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_273_order__eq__refl,axiom,
! [X3: real,Y: real] :
( ( X3 = Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_274_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_275_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_276_linorder__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_277_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_278_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_279_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_281_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_282_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_283_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_284_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_285_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_286_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_287_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_288_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_289_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_290_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_294_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_295_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_296_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_297_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_298_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_299_linorder__le__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_300_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_301_order__antisym__conv,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_302_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_303_order__antisym__conv,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_304_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_305_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_306_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_307_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_308_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_309_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_310_subset__singleton__iff,axiom,
! [X4: set_real,A2: real] :
( ( ord_less_eq_set_real @ X4 @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( ( X4 = bot_bot_set_real )
| ( X4
= ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_311_subset__singleton__iff,axiom,
! [X4: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_set_a )
| ( X4
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_312_subset__singleton__iff,axiom,
! [X4: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_313_subset__singleton__iff,axiom,
! [X4: set_int,A2: int] :
( ( ord_less_eq_set_int @ X4 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( ( X4 = bot_bot_set_int )
| ( X4
= ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_314_subset__singleton__iff,axiom,
! [X4: set_a,A2: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_315_subset__singletonD,axiom,
! [A: set_real,X3: real] :
( ( ord_less_eq_set_real @ A @ ( insert_real @ X3 @ bot_bot_set_real ) )
=> ( ( A = bot_bot_set_real )
| ( A
= ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_316_subset__singletonD,axiom,
! [A: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_317_subset__singletonD,axiom,
! [A: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_318_subset__singletonD,axiom,
! [A: set_int,X3: int] :
( ( ord_less_eq_set_int @ A @ ( insert_int @ X3 @ bot_bot_set_int ) )
=> ( ( A = bot_bot_set_int )
| ( A
= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_319_subset__singletonD,axiom,
! [A: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_320_singleton__inject,axiom,
! [A2: real,B2: real] :
( ( ( insert_real @ A2 @ bot_bot_set_real )
= ( insert_real @ B2 @ bot_bot_set_real ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_321_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_322_singleton__inject,axiom,
! [A2: set_a,B2: set_a] :
( ( ( insert_set_a @ A2 @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_323_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_324_singleton__inject,axiom,
! [A2: int,B2: int] :
( ( ( insert_int @ A2 @ bot_bot_set_int )
= ( insert_int @ B2 @ bot_bot_set_int ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_325_insert__not__empty,axiom,
! [A2: real,A: set_real] :
( ( insert_real @ A2 @ A )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_326_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_327_insert__not__empty,axiom,
! [A2: set_a,A: set_set_a] :
( ( insert_set_a @ A2 @ A )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_328_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_329_insert__not__empty,axiom,
! [A2: int,A: set_int] :
( ( insert_int @ A2 @ A )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_330_doubleton__eq__iff,axiom,
! [A2: real,B2: real,C: real,D2: real] :
( ( ( insert_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) )
= ( insert_real @ C @ ( insert_real @ D2 @ bot_bot_set_real ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_331_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_332_doubleton__eq__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ( insert_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_333_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_334_doubleton__eq__iff,axiom,
! [A2: int,B2: int,C: int,D2: int] :
( ( ( insert_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_335_singleton__iff,axiom,
! [B2: real,A2: real] :
( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_336_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_337_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_338_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_339_singleton__iff,axiom,
! [B2: int,A2: int] :
( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_340_singletonD,axiom,
! [B2: real,A2: real] :
( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_341_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_342_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_343_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_344_singletonD,axiom,
! [B2: int,A2: int] :
( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_345_finite_OinsertI,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( finite_finite_real @ ( insert_real @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_346_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_347_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_348_finite_OinsertI,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( insert_int @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_349_finite_OinsertI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_350_bot_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
=> ( A2 = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_351_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
=> ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_352_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_353_bot_Oextremum__uniqueI,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
=> ( A2 = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_354_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_355_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_356_bot_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_357_bot_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_358_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_359_bot_Oextremum__unique,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_360_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_361_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_362_bot_Oextremum,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% bot.extremum
thf(fact_363_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_364_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_365_bot_Oextremum,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% bot.extremum
thf(fact_366_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_367_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_368_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_369_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_370_finite__has__minimal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_371_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_372_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_373_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_374_finite__has__maximal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ A2 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_375_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A2 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_376_rev__finite__subset,axiom,
! [B: set_real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( finite_finite_real @ A ) ) ) ).
% rev_finite_subset
thf(fact_377_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_378_rev__finite__subset,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_379_rev__finite__subset,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_380_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_381_infinite__super,axiom,
! [S: set_real,T2: set_real] :
( ( ord_less_eq_set_real @ S @ T2 )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ T2 ) ) ) ).
% infinite_super
thf(fact_382_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_383_infinite__super,axiom,
! [S: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S @ T2 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_384_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_385_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_386_finite__subset,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( finite_finite_real @ B )
=> ( finite_finite_real @ A ) ) ) ).
% finite_subset
thf(fact_387_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_388_finite__subset,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( finite_finite_int @ B )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_389_finite__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_390_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_391_finite__subset__induct_H,axiom,
! [F2: set_real,A: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A4 @ A )
=> ( ( ord_less_eq_set_real @ F3 @ A )
=> ( ~ ( member_real @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_392_finite__subset__induct_H,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A4 @ A )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A )
=> ( ~ ( member_set_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_393_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_394_finite__subset__induct_H,axiom,
! [F2: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( ord_less_eq_set_int @ F2 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( member_int @ A4 @ A )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ~ ( member_int @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_395_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_396_finite__subset__induct,axiom,
! [F2: set_real,A: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A4 @ A )
=> ( ~ ( member_real @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_397_finite__subset__induct,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A4 @ A )
=> ( ~ ( member_set_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_398_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_399_finite__subset__induct,axiom,
! [F2: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( ord_less_eq_set_int @ F2 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( member_int @ A4 @ A )
=> ( ~ ( member_int @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_400_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_401_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X: real] : ( member_real @ X @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_402_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_403_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_404_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_405_ex__in__conv,axiom,
! [A: set_int] :
( ( ? [X: int] : ( member_int @ X @ A ) )
= ( A != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_406_equals0I,axiom,
! [A: set_real] :
( ! [Y4: real] :
~ ( member_real @ Y4 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_407_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_408_equals0I,axiom,
! [A: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_409_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_410_equals0I,axiom,
! [A: set_int] :
( ! [Y4: int] :
~ ( member_int @ Y4 @ A )
=> ( A = bot_bot_set_int ) ) ).
% equals0I
thf(fact_411_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A2 @ A ) ) ).
% equals0D
thf(fact_412_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_413_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_414_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_415_equals0D,axiom,
! [A: set_int,A2: int] :
( ( A = bot_bot_set_int )
=> ~ ( member_int @ A2 @ A ) ) ).
% equals0D
thf(fact_416_emptyE,axiom,
! [A2: real] :
~ ( member_real @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_417_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_418_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_419_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_420_emptyE,axiom,
! [A2: int] :
~ ( member_int @ A2 @ bot_bot_set_int ) ).
% emptyE
thf(fact_421_infinite__finite__induct,axiom,
! [P: set_real > $o,A: set_real] :
( ! [A3: set_real] :
( ~ ( finite_finite_real @ A3 )
=> ( P @ A3 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_422_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A3: set_a] :
( ~ ( finite_finite_a @ A3 )
=> ( P @ A3 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_423_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A: set_set_a] :
( ! [A3: set_set_a] :
( ~ ( finite_finite_set_a @ A3 )
=> ( P @ A3 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_424_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A3: set_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( P @ A3 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_425_infinite__finite__induct,axiom,
! [P: set_int > $o,A: set_int] :
( ! [A3: set_int] :
( ~ ( finite_finite_int @ A3 )
=> ( P @ A3 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ~ ( member_int @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_426_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X2: real] : ( P @ ( insert_real @ X2 @ bot_bot_set_real ) )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_427_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_428_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X2: set_a] : ( P @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_429_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_430_finite__ne__induct,axiom,
! [F2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( F2 != bot_bot_set_int )
=> ( ! [X2: int] : ( P @ ( insert_int @ X2 @ bot_bot_set_int ) )
=> ( ! [X2: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( F3 != bot_bot_set_int )
=> ( ~ ( member_int @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_431_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_432_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_433_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_434_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_435_finite__induct,axiom,
! [F2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ~ ( member_int @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_436_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A5: set_real] :
( ( A5 = bot_bot_set_real )
| ? [A6: set_real,B4: real] :
( ( A5
= ( insert_real @ B4 @ A6 ) )
& ( finite_finite_real @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_437_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A5: set_a] :
( ( A5 = bot_bot_set_a )
| ? [A6: set_a,B4: a] :
( ( A5
= ( insert_a @ B4 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_438_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A5: set_set_a] :
( ( A5 = bot_bot_set_set_a )
| ? [A6: set_set_a,B4: set_a] :
( ( A5
= ( insert_set_a @ B4 @ A6 ) )
& ( finite_finite_set_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_439_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A6: set_nat,B4: nat] :
( ( A5
= ( insert_nat @ B4 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_440_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A5: set_int] :
( ( A5 = bot_bot_set_int )
| ? [A6: set_int,B4: int] :
( ( A5
= ( insert_int @ B4 @ A6 ) )
& ( finite_finite_int @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_441_sumset__subset__insert_I2_J,axiom,
! [A: set_a,B: set_a,X3: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X3 @ A ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_442_sumset__subset__insert_I1_J,axiom,
! [A: set_a,B: set_a,X3: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X3 @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_443_sumset__mono,axiom,
! [A7: set_a,A: set_a,B7: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ( ( ord_less_eq_set_a @ B7 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ B7 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_mono
thf(fact_444_sumset__subset__carrier,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_445_Min_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_446_Min_Obounded__iff,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X3 @ ( lattic8718645017227715691in_int @ A ) )
= ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_int @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_447_Min_Obounded__iff,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X3 @ ( lattic3629708407755379051n_real @ A ) )
= ( ! [X: real] :
( ( member_real @ X @ A )
=> ( ord_less_eq_real @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_448_finite__sumset,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset
thf(fact_449_sumset__commute,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A ) ) ).
% sumset_commute
thf(fact_450_sumset__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ C2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C2 ) ) ) ).
% sumset_assoc
thf(fact_451_sumset_OsumsetI,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ B )
=> ( ( member_a @ B2 @ g )
=> ( member_a @ ( addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_452_sumset_Osimps,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= ( ? [A5: a,B4: a] :
( ( A2
= ( addition @ A5 @ B4 ) )
& ( member_a @ A5 @ A )
& ( member_a @ A5 @ g )
& ( member_a @ B4 @ B )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumset.simps
thf(fact_453_sumset_Ocases,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
=> ~ ! [A4: a,B3: a] :
( ( A2
= ( addition @ A4 @ B3 ) )
=> ( ( member_a @ A4 @ A )
=> ( ( member_a @ A4 @ g )
=> ( ( member_a @ B3 @ B )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_454_assms_I4_J,axiom,
a0 != bot_bot_set_a ).
% assms(4)
thf(fact_455_assms_I2_J,axiom,
finite_finite_a @ a0 ).
% assms(2)
thf(fact_456_card__le__sumset,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_457_Min__singleton,axiom,
! [X3: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Min_singleton
thf(fact_458_Min__singleton,axiom,
! [X3: int] :
( ( lattic8718645017227715691in_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
= X3 ) ).
% Min_singleton
thf(fact_459_Min__singleton,axiom,
! [X3: real] :
( ( lattic3629708407755379051n_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
= X3 ) ).
% Min_singleton
thf(fact_460_associative,axiom,
! [A2: a,B2: a,C: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ g )
=> ( ( member_a @ C @ g )
=> ( ( addition @ ( addition @ A2 @ B2 ) @ C )
= ( addition @ A2 @ ( addition @ B2 @ C ) ) ) ) ) ) ).
% associative
thf(fact_461_composition__closed,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ g )
=> ( member_a @ ( addition @ A2 @ B2 ) @ g ) ) ) ).
% composition_closed
thf(fact_462_card__sumset__le,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ).
% card_sumset_le
thf(fact_463_sumset__empty_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_464_sumset__empty_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_465_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_466_Min__le,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 ) ) ) ).
% Min_le
thf(fact_467_Min__le,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ X3 ) ) ) ).
% Min_le
thf(fact_468_Min__le,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A ) @ X3 ) ) ) ).
% Min_le
thf(fact_469_Min__eqI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ord_less_eq_nat @ X3 @ Y4 ) )
=> ( ( member_nat @ X3 @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_470_Min__eqI,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ! [Y4: int] :
( ( member_int @ Y4 @ A )
=> ( ord_less_eq_int @ X3 @ Y4 ) )
=> ( ( member_int @ X3 @ A )
=> ( ( lattic8718645017227715691in_int @ A )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_471_Min__eqI,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ! [Y4: real] :
( ( member_real @ Y4 @ A )
=> ( ord_less_eq_real @ X3 @ Y4 ) )
=> ( ( member_real @ X3 @ A )
=> ( ( lattic3629708407755379051n_real @ A )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_472_Min_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_473_Min_OcoboundedI,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_474_Min_OcoboundedI,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_475_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_476_Min__in,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( member_int @ ( lattic8718645017227715691in_int @ A ) @ A ) ) ) ).
% Min_in
thf(fact_477_Min__in,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( member_real @ ( lattic3629708407755379051n_real @ A ) @ A ) ) ) ).
% Min_in
thf(fact_478_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y5: real] :
( ( member_real @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_479_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_480_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_481_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_int @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_482_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y5: real] :
( ( member_real @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_483_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_484_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_485_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F: int > real] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_int @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_486_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > int] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y5: real] :
( ( member_real @ Y5 @ S2 )
=> ( ord_less_eq_int @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_487_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > int] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_int @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_488_Min__eq__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M )
= ( ( member_nat @ M @ A )
& ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_489_Min__eq__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ( lattic8718645017227715691in_int @ A )
= M )
= ( ( member_int @ M @ A )
& ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_int @ M @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_490_Min__eq__iff,axiom,
! [A: set_real,M: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ( lattic3629708407755379051n_real @ A )
= M )
= ( ( member_real @ M @ A )
& ! [X: real] :
( ( member_real @ X @ A )
=> ( ord_less_eq_real @ M @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_491_Min__le__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_492_Min__le__iff,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ X3 )
= ( ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_eq_int @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_493_Min__le__iff,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A ) @ X3 )
= ( ? [X: real] :
( ( member_real @ X @ A )
& ( ord_less_eq_real @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_494_eq__Min__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat @ M @ A )
& ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_495_eq__Min__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( M
= ( lattic8718645017227715691in_int @ A ) )
= ( ( member_int @ M @ A )
& ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_int @ M @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_496_eq__Min__iff,axiom,
! [A: set_real,M: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( M
= ( lattic3629708407755379051n_real @ A ) )
= ( ( member_real @ M @ A )
& ! [X: real] :
( ( member_real @ X @ A )
=> ( ord_less_eq_real @ M @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_497_Min_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ X3 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_498_Min_OboundedE,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X3 @ ( lattic8718645017227715691in_int @ A ) )
=> ! [A8: int] :
( ( member_int @ A8 @ A )
=> ( ord_less_eq_int @ X3 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_499_Min_OboundedE,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X3 @ ( lattic3629708407755379051n_real @ A ) )
=> ! [A8: real] :
( ( member_real @ A8 @ A )
=> ( ord_less_eq_real @ X3 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_500_Min_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X3 @ A4 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_501_Min_OboundedI,axiom,
! [A: set_int,X3: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ X3 @ A4 ) )
=> ( ord_less_eq_int @ X3 @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_502_Min_OboundedI,axiom,
! [A: set_real,X3: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ord_less_eq_real @ X3 @ A4 ) )
=> ( ord_less_eq_real @ X3 @ ( lattic3629708407755379051n_real @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_503_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ A )
=> ( ord_less_eq_nat @ A2 @ B3 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_504_Min__insert2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ! [B3: int] :
( ( member_int @ B3 @ A )
=> ( ord_less_eq_int @ A2 @ B3 ) )
=> ( ( lattic8718645017227715691in_int @ ( insert_int @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_505_Min__insert2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ! [B3: real] :
( ( member_real @ B3 @ A )
=> ( ord_less_eq_real @ A2 @ B3 ) )
=> ( ( lattic3629708407755379051n_real @ ( insert_real @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_506_Min__antimono,axiom,
! [M2: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_507_Min__antimono,axiom,
! [M2: set_int,N: set_int] :
( ( ord_less_eq_set_int @ M2 @ N )
=> ( ( M2 != bot_bot_set_int )
=> ( ( finite_finite_int @ N )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N ) @ ( lattic8718645017227715691in_int @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_508_Min__antimono,axiom,
! [M2: set_real,N: set_real] :
( ( ord_less_eq_set_real @ M2 @ N )
=> ( ( M2 != bot_bot_set_real )
=> ( ( finite_finite_real @ N )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ N ) @ ( lattic3629708407755379051n_real @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_509_Min_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_510_Min_Osubset__imp,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B ) @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_511_Min_Osubset__imp,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( A != bot_bot_set_real )
=> ( ( finite_finite_real @ B )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ B ) @ ( lattic3629708407755379051n_real @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_512_sumsetp_OsumsetI,axiom,
! [A: a > $o,A2: a,B: a > $o,B2: a] :
( ( A @ A2 )
=> ( ( member_a @ A2 @ g )
=> ( ( B @ B2 )
=> ( ( member_a @ B2 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ ( addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_513_sumsetp_Osimps,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
= ( ? [A5: a,B4: a] :
( ( A2
= ( addition @ A5 @ B4 ) )
& ( A @ A5 )
& ( member_a @ A5 @ g )
& ( B @ B4 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_514_sumsetp_Ocases,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
=> ~ ! [A4: a,B3: a] :
( ( A2
= ( addition @ A4 @ B3 ) )
=> ( ( A @ A4 )
=> ( ( member_a @ A4 @ g )
=> ( ( B @ B3 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_515_finite__Diff__insert,axiom,
! [A: set_real,A2: real,B: set_real] :
( ( finite_finite_real @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) ) )
= ( finite_finite_real @ ( minus_minus_set_real @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_516_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_517_finite__Diff__insert,axiom,
! [A: set_int,A2: int,B: set_int] :
( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) ) )
= ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_518_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_519_finite__Diff__insert,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_520_insert__Diff__single,axiom,
! [A2: real,A: set_real] :
( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
= ( insert_real @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_521_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_522_insert__Diff__single,axiom,
! [A2: int,A: set_int] :
( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( insert_int @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_523_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_524_insert__Diff__single,axiom,
! [A2: set_a,A: set_set_a] :
( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_525_Diff__eq__empty__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( minus_minus_set_real @ A @ B )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_526_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_527_Diff__eq__empty__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( minus_minus_set_int @ A @ B )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_528_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_529_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_530_sumsetdiff__sing,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_531_Pow__iff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_532_DiffI,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ A )
=> ( ~ ( member_real @ C @ B )
=> ( member_real @ C @ ( minus_minus_set_real @ A @ B ) ) ) ) ).
% DiffI
thf(fact_533_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_534_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_535_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_536_Diff__iff,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
= ( ( member_real @ C @ A )
& ~ ( member_real @ C @ B ) ) ) ).
% Diff_iff
thf(fact_537_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_538_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_539_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_540_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_541_Diff__idemp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_542_Diff__cancel,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ A )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_543_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_544_Diff__cancel,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ A @ A )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_545_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_546_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_547_empty__Diff,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_548_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_549_empty__Diff,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_550_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_551_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_552_Diff__empty,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ bot_bot_set_real )
= A ) ).
% Diff_empty
thf(fact_553_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_554_Diff__empty,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ A @ bot_bot_set_int )
= A ) ).
% Diff_empty
thf(fact_555_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_556_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_557_finite__Diff2,axiom,
! [B: set_real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( finite_finite_real @ ( minus_minus_set_real @ A @ B ) )
= ( finite_finite_real @ A ) ) ) ).
% finite_Diff2
thf(fact_558_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_559_finite__Diff2,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_Diff2
thf(fact_560_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_561_finite__Diff2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( finite_finite_set_a @ A ) ) ) ).
% finite_Diff2
thf(fact_562_finite__Diff,axiom,
! [A: set_real,B: set_real] :
( ( finite_finite_real @ A )
=> ( finite_finite_real @ ( minus_minus_set_real @ A @ B ) ) ) ).
% finite_Diff
thf(fact_563_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_564_finite__Diff,axiom,
! [A: set_int,B: set_int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).
% finite_Diff
thf(fact_565_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_566_finite__Diff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_567_Diff__insert0,axiom,
! [X3: real,A: set_real,B: set_real] :
( ~ ( member_real @ X3 @ A )
=> ( ( minus_minus_set_real @ A @ ( insert_real @ X3 @ B ) )
= ( minus_minus_set_real @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_568_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_569_Diff__insert0,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_570_Diff__insert0,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_571_insert__Diff1,axiom,
! [X3: real,B: set_real,A: set_real] :
( ( member_real @ X3 @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X3 @ A ) @ B )
= ( minus_minus_set_real @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_572_insert__Diff1,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_573_insert__Diff1,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_574_insert__Diff1,axiom,
! [X3: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_575_Pow__singleton__iff,axiom,
! [X4: set_real,Y6: set_real] :
( ( ( pow_real @ X4 )
= ( insert_set_real @ Y6 @ bot_bot_set_set_real ) )
= ( ( X4 = bot_bot_set_real )
& ( Y6 = bot_bot_set_real ) ) ) ).
% Pow_singleton_iff
thf(fact_576_Pow__singleton__iff,axiom,
! [X4: set_set_a,Y6: set_set_a] :
( ( ( pow_set_a @ X4 )
= ( insert_set_set_a @ Y6 @ bot_bo3380559777022489994_set_a ) )
= ( ( X4 = bot_bot_set_set_a )
& ( Y6 = bot_bot_set_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_577_Pow__singleton__iff,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y6 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_578_Pow__singleton__iff,axiom,
! [X4: set_int,Y6: set_int] :
( ( ( pow_int @ X4 )
= ( insert_set_int @ Y6 @ bot_bot_set_set_int ) )
= ( ( X4 = bot_bot_set_int )
& ( Y6 = bot_bot_set_int ) ) ) ).
% Pow_singleton_iff
thf(fact_579_Pow__singleton__iff,axiom,
! [X4: set_a,Y6: set_a] :
( ( ( pow_a @ X4 )
= ( insert_set_a @ Y6 @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_a )
& ( Y6 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_580_Pow__empty,axiom,
( ( pow_real @ bot_bot_set_real )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).
% Pow_empty
thf(fact_581_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_582_Pow__empty,axiom,
( ( pow_set_a @ bot_bot_set_set_a )
= ( insert_set_set_a @ bot_bot_set_set_a @ bot_bo3380559777022489994_set_a ) ) ).
% Pow_empty
thf(fact_583_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_584_Pow__empty,axiom,
( ( pow_int @ bot_bot_set_int )
= ( insert_set_int @ bot_bot_set_int @ bot_bot_set_set_int ) ) ).
% Pow_empty
thf(fact_585_finite__Pow__iff,axiom,
! [A: set_real] :
( ( finite9007344921179782393t_real @ ( pow_real @ A ) )
= ( finite_finite_real @ A ) ) ).
% finite_Pow_iff
thf(fact_586_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_587_finite__Pow__iff,axiom,
! [A: set_int] :
( ( finite6197958912794628473et_int @ ( pow_int @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_Pow_iff
thf(fact_588_finite__Pow__iff,axiom,
! [A: set_set_a] :
( ( finite7209287970140883943_set_a @ ( pow_set_a @ A ) )
= ( finite_finite_set_a @ A ) ) ).
% finite_Pow_iff
thf(fact_589_finite__Pow__iff,axiom,
! [A: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_Pow_iff
thf(fact_590_PowI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_591_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_592_DiffE,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% DiffE
thf(fact_593_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_594_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_595_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_596_DiffD1,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ( member_real @ C @ A ) ) ).
% DiffD1
thf(fact_597_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_598_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_599_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_600_DiffD2,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( member_real @ C @ B ) ) ).
% DiffD2
thf(fact_601_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_602_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_603_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_604_Pow__top,axiom,
! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% Pow_top
thf(fact_605_Pow__not__empty,axiom,
! [A: set_a] :
( ( pow_a @ A )
!= bot_bot_set_set_a ) ).
% Pow_not_empty
thf(fact_606_card__le__sym__Diff,axiom,
! [A: set_real,B: set_real] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_real @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A ) @ ( finite_card_real @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A @ B ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_607_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_608_card__le__sym__Diff,axiom,
! [A: set_int,B: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_609_card__le__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_610_card__le__sym__Diff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_611_card__Diff1__le,axiom,
! [A: set_real,X3: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A ) ) ).
% card_Diff1_le
thf(fact_612_card__Diff1__le,axiom,
! [A: set_nat,X3: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% card_Diff1_le
thf(fact_613_card__Diff1__le,axiom,
! [A: set_int,X3: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ).
% card_Diff1_le
thf(fact_614_card__Diff1__le,axiom,
! [A: set_a,X3: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ).
% card_Diff1_le
thf(fact_615_card__Diff1__le,axiom,
! [A: set_set_a,X3: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A ) ) ).
% card_Diff1_le
thf(fact_616_card__insert__le,axiom,
! [A: set_set_a,X3: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ ( insert_set_a @ X3 @ A ) ) ) ).
% card_insert_le
thf(fact_617_card__insert__le,axiom,
! [A: set_a,X3: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X3 @ A ) ) ) ).
% card_insert_le
thf(fact_618_Diff__infinite__finite,axiom,
! [T2: set_real,S: set_real] :
( ( finite_finite_real @ T2 )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_619_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_620_Diff__infinite__finite,axiom,
! [T2: set_int,S: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_621_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_622_Diff__infinite__finite,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_623_Diff__mono,axiom,
! [A: set_set_a,C2: set_set_a,D: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_624_Diff__mono,axiom,
! [A: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_625_Diff__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_626_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_627_double__diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_628_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_629_insert__Diff__if,axiom,
! [X3: real,B: set_real,A: set_real] :
( ( ( member_real @ X3 @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X3 @ A ) @ B )
= ( minus_minus_set_real @ A @ B ) ) )
& ( ~ ( member_real @ X3 @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X3 @ A ) @ B )
= ( insert_real @ X3 @ ( minus_minus_set_real @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_630_insert__Diff__if,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_631_insert__Diff__if,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_632_insert__Diff__if,axiom,
! [X3: set_a,B: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) )
& ( ~ ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( insert_set_a @ X3 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_633_Pow__bottom,axiom,
! [B: set_real] : ( member_set_real @ bot_bot_set_real @ ( pow_real @ B ) ) ).
% Pow_bottom
thf(fact_634_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_635_Pow__bottom,axiom,
! [B: set_set_a] : ( member_set_set_a @ bot_bot_set_set_a @ ( pow_set_a @ B ) ) ).
% Pow_bottom
thf(fact_636_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_637_Pow__bottom,axiom,
! [B: set_int] : ( member_set_int @ bot_bot_set_int @ ( pow_int @ B ) ) ).
% Pow_bottom
thf(fact_638_PowD,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% PowD
thf(fact_639_Pow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_640_card__mono,axiom,
! [B: set_real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_real @ A ) @ ( finite_card_real @ B ) ) ) ) ).
% card_mono
thf(fact_641_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_642_card__mono,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) ) ) ) ).
% card_mono
thf(fact_643_card__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B ) ) ) ) ).
% card_mono
thf(fact_644_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_645_card__seteq,axiom,
! [B: set_real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_real @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_646_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_647_card__seteq,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B ) @ ( finite_card_int @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_648_card__seteq,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_set_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_649_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_650_exists__subset__between,axiom,
! [A: set_real,N2: nat,C2: set_real] :
( ( ord_less_eq_nat @ ( finite_card_real @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_real @ C2 ) )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( finite_finite_real @ C2 )
=> ? [B6: set_real] :
( ( ord_less_eq_set_real @ A @ B6 )
& ( ord_less_eq_set_real @ B6 @ C2 )
& ( ( finite_card_real @ B6 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_651_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_652_exists__subset__between,axiom,
! [A: set_int,N2: nat,C2: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ C2 ) )
=> ( ( ord_less_eq_set_int @ A @ C2 )
=> ( ( finite_finite_int @ C2 )
=> ? [B6: set_int] :
( ( ord_less_eq_set_int @ A @ B6 )
& ( ord_less_eq_set_int @ B6 @ C2 )
& ( ( finite_card_int @ B6 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_653_exists__subset__between,axiom,
! [A: set_set_a,N2: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ C2 )
& ( ( finite_card_set_a @ B6 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_654_exists__subset__between,axiom,
! [A: set_a,N2: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A @ B6 )
& ( ord_less_eq_set_a @ B6 @ C2 )
& ( ( finite_card_a @ B6 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_655_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_real] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_real @ S ) )
=> ~ ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
=> ( ( ( finite_card_real @ T3 )
= N2 )
=> ~ ( finite_finite_real @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_656_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_657_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_int] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S ) )
=> ~ ! [T3: set_int] :
( ( ord_less_eq_set_int @ T3 @ S )
=> ( ( ( finite_card_int @ T3 )
= N2 )
=> ~ ( finite_finite_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_658_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N2 )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_659_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N2 )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_660_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_real,C2: nat] :
( ! [G: set_real] :
( ( ord_less_eq_set_real @ G @ F2 )
=> ( ( finite_finite_real @ G )
=> ( ord_less_eq_nat @ ( finite_card_real @ G ) @ C2 ) ) )
=> ( ( finite_finite_real @ F2 )
& ( ord_less_eq_nat @ ( finite_card_real @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_661_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_662_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_int,C2: nat] :
( ! [G: set_int] :
( ( ord_less_eq_set_int @ G @ F2 )
=> ( ( finite_finite_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C2 ) ) )
=> ( ( finite_finite_int @ F2 )
& ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_663_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_664_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_665_infinite__arbitrarily__large,axiom,
! [A: set_real,N2: nat] :
( ~ ( finite_finite_real @ A )
=> ? [B6: set_real] :
( ( finite_finite_real @ B6 )
& ( ( finite_card_real @ B6 )
= N2 )
& ( ord_less_eq_set_real @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_666_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N2 )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_667_infinite__arbitrarily__large,axiom,
! [A: set_int,N2: nat] :
( ~ ( finite_finite_int @ A )
=> ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ( finite_card_int @ B6 )
= N2 )
& ( ord_less_eq_set_int @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_668_infinite__arbitrarily__large,axiom,
! [A: set_set_a,N2: nat] :
( ~ ( finite_finite_set_a @ A )
=> ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ( finite_card_set_a @ B6 )
= N2 )
& ( ord_le3724670747650509150_set_a @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_669_infinite__arbitrarily__large,axiom,
! [A: set_a,N2: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N2 )
& ( ord_less_eq_set_a @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_670_card__subset__eq,axiom,
! [B: set_real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( ( finite_card_real @ A )
= ( finite_card_real @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_671_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_672_card__subset__eq,axiom,
! [B: set_int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_673_card__subset__eq,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( finite_card_set_a @ A )
= ( finite_card_set_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_674_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_675_Diff__insert__absorb,axiom,
! [X3: real,A: set_real] :
( ~ ( member_real @ X3 @ A )
=> ( ( minus_minus_set_real @ ( insert_real @ X3 @ A ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_676_Diff__insert__absorb,axiom,
! [X3: nat,A: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_677_Diff__insert__absorb,axiom,
! [X3: int,A: set_int] :
( ~ ( member_int @ X3 @ A )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_678_Diff__insert__absorb,axiom,
! [X3: a,A: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_679_Diff__insert__absorb,axiom,
! [X3: set_a,A: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_680_Diff__insert2,axiom,
! [A: set_real,A2: real,B: set_real] :
( ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) @ B ) ) ).
% Diff_insert2
thf(fact_681_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_682_Diff__insert2,axiom,
! [A: set_int,A2: int,B: set_int] :
( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B ) ) ).
% Diff_insert2
thf(fact_683_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_684_Diff__insert2,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_685_insert__Diff,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
= A ) ) ).
% insert_Diff
thf(fact_686_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_687_insert__Diff,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= A ) ) ).
% insert_Diff
thf(fact_688_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_689_insert__Diff,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_690_Diff__insert,axiom,
! [A: set_real,A2: real,B: set_real] :
( ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A @ B ) @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_691_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_692_Diff__insert,axiom,
! [A: set_int,A2: int,B: set_int] :
( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_693_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_694_Diff__insert,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_695_subset__Diff__insert,axiom,
! [A: set_real,B: set_real,X3: real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ ( insert_real @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ C2 ) )
& ~ ( member_real @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_696_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X3: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_697_subset__Diff__insert,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X3 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ C2 ) )
& ~ ( member_set_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_698_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X3: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_699_finite__empty__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ A )
=> ( ! [A4: real,A3: set_real] :
( ( finite_finite_real @ A3 )
=> ( ( member_real @ A4 @ A3 )
=> ( ( P @ A3 )
=> ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_700_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A4 @ A3 )
=> ( ( P @ A3 )
=> ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_701_finite__empty__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ A )
=> ( ! [A4: int,A3: set_int] :
( ( finite_finite_int @ A3 )
=> ( ( member_int @ A4 @ A3 )
=> ( ( P @ A3 )
=> ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_702_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A4: a,A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( member_a @ A4 @ A3 )
=> ( ( P @ A3 )
=> ( P @ ( minus_minus_set_a @ A3 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_703_finite__empty__induct,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ( P @ A )
=> ( ! [A4: set_a,A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A4 @ A3 )
=> ( ( P @ A3 )
=> ( P @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_704_infinite__coinduct,axiom,
! [X4: set_real > $o,A: set_real] :
( ( X4 @ A )
=> ( ! [A3: set_real] :
( ( X4 @ A3 )
=> ? [X5: real] :
( ( member_real @ X5 @ A3 )
& ( ( X4 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X5 @ bot_bot_set_real ) ) )
| ~ ( finite_finite_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ) )
=> ~ ( finite_finite_real @ A ) ) ) ).
% infinite_coinduct
thf(fact_705_infinite__coinduct,axiom,
! [X4: set_nat > $o,A: set_nat] :
( ( X4 @ A )
=> ( ! [A3: set_nat] :
( ( X4 @ A3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A3 )
& ( ( X4 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_706_infinite__coinduct,axiom,
! [X4: set_int > $o,A: set_int] :
( ( X4 @ A )
=> ( ! [A3: set_int] :
( ( X4 @ A3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A3 )
& ( ( X4 @ ( minus_minus_set_int @ A3 @ ( insert_int @ X5 @ bot_bot_set_int ) ) )
| ~ ( finite_finite_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) ) ) )
=> ~ ( finite_finite_int @ A ) ) ) ).
% infinite_coinduct
thf(fact_707_infinite__coinduct,axiom,
! [X4: set_a > $o,A: set_a] :
( ( X4 @ A )
=> ( ! [A3: set_a] :
( ( X4 @ A3 )
=> ? [X5: a] :
( ( member_a @ X5 @ A3 )
& ( ( X4 @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_708_infinite__coinduct,axiom,
! [X4: set_set_a > $o,A: set_set_a] :
( ( X4 @ A )
=> ( ! [A3: set_set_a] :
( ( X4 @ A3 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
& ( ( X4 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_709_infinite__remove,axiom,
! [S: set_real,A2: real] :
( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% infinite_remove
thf(fact_710_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_711_infinite__remove,axiom,
! [S: set_int,A2: int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% infinite_remove
thf(fact_712_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_713_infinite__remove,axiom,
! [S: set_set_a,A2: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_714_Diff__single__insert,axiom,
! [A: set_real,X3: real,B: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B )
=> ( ord_less_eq_set_real @ A @ ( insert_real @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_715_Diff__single__insert,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_716_Diff__single__insert,axiom,
! [A: set_int,X3: int,B: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B )
=> ( ord_less_eq_set_int @ A @ ( insert_int @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_717_Diff__single__insert,axiom,
! [A: set_set_a,X3: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_718_Diff__single__insert,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_719_subset__insert__iff,axiom,
! [A: set_real,X3: real,B: set_real] :
( ( ord_less_eq_set_real @ A @ ( insert_real @ X3 @ B ) )
= ( ( ( member_real @ X3 @ A )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B ) )
& ( ~ ( member_real @ X3 @ A )
=> ( ord_less_eq_set_real @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_720_subset__insert__iff,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_721_subset__insert__iff,axiom,
! [A: set_int,X3: int,B: set_int] :
( ( ord_less_eq_set_int @ A @ ( insert_int @ X3 @ B ) )
= ( ( ( member_int @ X3 @ A )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B ) )
& ( ~ ( member_int @ X3 @ A )
=> ( ord_less_eq_set_int @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_722_subset__insert__iff,axiom,
! [A: set_set_a,X3: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ( ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_723_subset__insert__iff,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_724_finite__remove__induct,axiom,
! [B: set_real,P: set_real > $o] :
( ( finite_finite_real @ B )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A3: set_real] :
( ( finite_finite_real @ A3 )
=> ( ( A3 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A3 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A3 )
=> ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_725_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_726_finite__remove__induct,axiom,
! [B: set_int,P: set_int > $o] :
( ( finite_finite_int @ B )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A3: set_int] :
( ( finite_finite_int @ A3 )
=> ( ( A3 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A3 @ B )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A3 )
=> ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_727_finite__remove__induct,axiom,
! [B: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( P @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_728_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( P @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_729_remove__induct,axiom,
! [P: set_real > $o,B: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B )
=> ( P @ B ) )
=> ( ! [A3: set_real] :
( ( finite_finite_real @ A3 )
=> ( ( A3 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A3 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A3 )
=> ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_730_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_731_remove__induct,axiom,
! [P: set_int > $o,B: set_int] :
( ( P @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B )
=> ( P @ B ) )
=> ( ! [A3: set_int] :
( ( finite_finite_int @ A3 )
=> ( ( A3 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A3 @ B )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A3 )
=> ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_732_remove__induct,axiom,
! [P: set_set_a > $o,B: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B )
=> ( P @ B ) )
=> ( ! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ( P @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_733_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( P @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A3 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_734_K0,axiom,
ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a0 @ b ) ) ) @ ( times_times_real @ k0 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a0 ) ) ) ).
% K0
thf(fact_735_card__sumset__0__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_736_infinite__sumset__aux,axiom,
! [A: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_737_infinite__sumset__iff,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_738_surj__card__le,axiom,
! [A: set_a,B: set_set_a,F: a > set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_739_surj__card__le,axiom,
! [A: set_set_a,B: set_real,F: set_a > real] :
( ( finite_finite_set_a @ A )
=> ( ( ord_less_eq_set_real @ B @ ( image_set_a_real @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_set_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_740_surj__card__le,axiom,
! [A: set_real,B: set_a,F: real > a] :
( ( finite_finite_real @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_real_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_real @ A ) ) ) ) ).
% surj_card_le
thf(fact_741_surj__card__le,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_742_surj__card__le,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_743_surj__card__le,axiom,
! [A: set_int,B: set_a,F: int > a] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_int_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_int @ A ) ) ) ) ).
% surj_card_le
thf(fact_744_surj__card__le,axiom,
! [A: set_set_a,B: set_a,F: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_set_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_745_finite__sumset_H,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset'
thf(fact_746_sumset__empty_H_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_747_sumset__empty_H_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_748_image__eqI,axiom,
! [B2: a,F: a > a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_749_image__eqI,axiom,
! [B2: real,F: a > real,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_real @ B2 @ ( image_a_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_750_image__eqI,axiom,
! [B2: nat,F: a > nat,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_751_image__eqI,axiom,
! [B2: a,F: real > a,X3: real,A: set_real] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_real @ X3 @ A )
=> ( member_a @ B2 @ ( image_real_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_752_image__eqI,axiom,
! [B2: real,F: real > real,X3: real,A: set_real] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_real @ X3 @ A )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_753_image__eqI,axiom,
! [B2: nat,F: real > nat,X3: real,A: set_real] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_real @ X3 @ A )
=> ( member_nat @ B2 @ ( image_real_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_754_image__eqI,axiom,
! [B2: a,F: nat > a,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_755_image__eqI,axiom,
! [B2: real,F: nat > real,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_756_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_757_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_758_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_759_IntI,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ A )
=> ( ( member_real @ C @ B )
=> ( member_real @ C @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% IntI
thf(fact_760_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_761_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_762_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_763_Int__iff,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A @ B ) )
= ( ( member_real @ C @ A )
& ( member_real @ C @ B ) ) ) ).
% Int_iff
thf(fact_764_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_765_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_766_card__sumset__0__iff_H,axiom,
! [A: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_767_image__empty,axiom,
! [F: real > real] :
( ( image_real_real @ F @ bot_bot_set_real )
= bot_bot_set_real ) ).
% image_empty
thf(fact_768_image__empty,axiom,
! [F: real > a] :
( ( image_real_a @ F @ bot_bot_set_real )
= bot_bot_set_a ) ).
% image_empty
thf(fact_769_image__empty,axiom,
! [F: real > nat] :
( ( image_real_nat @ F @ bot_bot_set_real )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_770_image__empty,axiom,
! [F: real > int] :
( ( image_real_int @ F @ bot_bot_set_real )
= bot_bot_set_int ) ).
% image_empty
thf(fact_771_image__empty,axiom,
! [F: a > real] :
( ( image_a_real @ F @ bot_bot_set_a )
= bot_bot_set_real ) ).
% image_empty
thf(fact_772_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_773_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_774_image__empty,axiom,
! [F: a > int] :
( ( image_a_int @ F @ bot_bot_set_a )
= bot_bot_set_int ) ).
% image_empty
thf(fact_775_image__empty,axiom,
! [F: nat > real] :
( ( image_nat_real @ F @ bot_bot_set_nat )
= bot_bot_set_real ) ).
% image_empty
thf(fact_776_image__empty,axiom,
! [F: nat > a] :
( ( image_nat_a @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_777_empty__is__image,axiom,
! [F: real > real,A: set_real] :
( ( bot_bot_set_real
= ( image_real_real @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_778_empty__is__image,axiom,
! [F: a > real,A: set_a] :
( ( bot_bot_set_real
= ( image_a_real @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_779_empty__is__image,axiom,
! [F: nat > real,A: set_nat] :
( ( bot_bot_set_real
= ( image_nat_real @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_780_empty__is__image,axiom,
! [F: int > real,A: set_int] :
( ( bot_bot_set_real
= ( image_int_real @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_781_empty__is__image,axiom,
! [F: real > a,A: set_real] :
( ( bot_bot_set_a
= ( image_real_a @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_782_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_783_empty__is__image,axiom,
! [F: nat > a,A: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_784_empty__is__image,axiom,
! [F: int > a,A: set_int] :
( ( bot_bot_set_a
= ( image_int_a @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_785_empty__is__image,axiom,
! [F: real > nat,A: set_real] :
( ( bot_bot_set_nat
= ( image_real_nat @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_786_empty__is__image,axiom,
! [F: a > nat,A: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_787_image__is__empty,axiom,
! [F: real > real,A: set_real] :
( ( ( image_real_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_788_image__is__empty,axiom,
! [F: a > real,A: set_a] :
( ( ( image_a_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_789_image__is__empty,axiom,
! [F: nat > real,A: set_nat] :
( ( ( image_nat_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_790_image__is__empty,axiom,
! [F: int > real,A: set_int] :
( ( ( image_int_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_791_image__is__empty,axiom,
! [F: real > a,A: set_real] :
( ( ( image_real_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_792_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_793_image__is__empty,axiom,
! [F: nat > a,A: set_nat] :
( ( ( image_nat_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_794_image__is__empty,axiom,
! [F: int > a,A: set_int] :
( ( ( image_int_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_795_image__is__empty,axiom,
! [F: real > nat,A: set_real] :
( ( ( image_real_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_796_image__is__empty,axiom,
! [F: a > nat,A: set_a] :
( ( ( image_a_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_797_finite__imageI,axiom,
! [F2: set_real,H: real > real] :
( ( finite_finite_real @ F2 )
=> ( finite_finite_real @ ( image_real_real @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_798_finite__imageI,axiom,
! [F2: set_real,H: real > a] :
( ( finite_finite_real @ F2 )
=> ( finite_finite_a @ ( image_real_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_799_finite__imageI,axiom,
! [F2: set_real,H: real > nat] :
( ( finite_finite_real @ F2 )
=> ( finite_finite_nat @ ( image_real_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_800_finite__imageI,axiom,
! [F2: set_real,H: real > int] :
( ( finite_finite_real @ F2 )
=> ( finite_finite_int @ ( image_real_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_801_finite__imageI,axiom,
! [F2: set_a,H: a > real] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_real @ ( image_a_real @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_802_finite__imageI,axiom,
! [F2: set_a,H: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_803_finite__imageI,axiom,
! [F2: set_a,H: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_804_finite__imageI,axiom,
! [F2: set_a,H: a > int] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_int @ ( image_a_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_805_finite__imageI,axiom,
! [F2: set_nat,H: nat > real] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_real @ ( image_nat_real @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_806_finite__imageI,axiom,
! [F2: set_nat,H: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_807_insert__image,axiom,
! [X3: a,A: set_a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( ( insert_a @ ( F @ X3 ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_808_insert__image,axiom,
! [X3: a,A: set_a,F: a > set_a] :
( ( member_a @ X3 @ A )
=> ( ( insert_set_a @ ( F @ X3 ) @ ( image_a_set_a @ F @ A ) )
= ( image_a_set_a @ F @ A ) ) ) ).
% insert_image
thf(fact_809_insert__image,axiom,
! [X3: set_a,A: set_set_a,F: set_a > real] :
( ( member_set_a @ X3 @ A )
=> ( ( insert_real @ ( F @ X3 ) @ ( image_set_a_real @ F @ A ) )
= ( image_set_a_real @ F @ A ) ) ) ).
% insert_image
thf(fact_810_insert__image,axiom,
! [X3: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X3 @ A )
=> ( ( insert_a @ ( F @ X3 ) @ ( image_set_a_a @ F @ A ) )
= ( image_set_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_811_insert__image,axiom,
! [X3: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( insert_set_a @ ( F @ X3 ) @ ( image_set_a_set_a @ F @ A ) )
= ( image_set_a_set_a @ F @ A ) ) ) ).
% insert_image
thf(fact_812_insert__image,axiom,
! [X3: real,A: set_real,F: real > a] :
( ( member_real @ X3 @ A )
=> ( ( insert_a @ ( F @ X3 ) @ ( image_real_a @ F @ A ) )
= ( image_real_a @ F @ A ) ) ) ).
% insert_image
thf(fact_813_insert__image,axiom,
! [X3: real,A: set_real,F: real > set_a] :
( ( member_real @ X3 @ A )
=> ( ( insert_set_a @ ( F @ X3 ) @ ( image_real_set_a @ F @ A ) )
= ( image_real_set_a @ F @ A ) ) ) ).
% insert_image
thf(fact_814_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( ( insert_a @ ( F @ X3 ) @ ( image_nat_a @ F @ A ) )
= ( image_nat_a @ F @ A ) ) ) ).
% insert_image
thf(fact_815_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > set_a] :
( ( member_nat @ X3 @ A )
=> ( ( insert_set_a @ ( F @ X3 ) @ ( image_nat_set_a @ F @ A ) )
= ( image_nat_set_a @ F @ A ) ) ) ).
% insert_image
thf(fact_816_image__insert,axiom,
! [F: a > a,A2: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A2 @ B ) )
= ( insert_a @ ( F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_817_image__insert,axiom,
! [F: a > set_a,A2: a,B: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A2 @ B ) )
= ( insert_set_a @ ( F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_insert
thf(fact_818_image__insert,axiom,
! [F: set_a > real,A2: set_a,B: set_set_a] :
( ( image_set_a_real @ F @ ( insert_set_a @ A2 @ B ) )
= ( insert_real @ ( F @ A2 ) @ ( image_set_a_real @ F @ B ) ) ) ).
% image_insert
thf(fact_819_image__insert,axiom,
! [F: set_a > a,A2: set_a,B: set_set_a] :
( ( image_set_a_a @ F @ ( insert_set_a @ A2 @ B ) )
= ( insert_a @ ( F @ A2 ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_820_image__insert,axiom,
! [F: set_a > set_a,A2: set_a,B: set_set_a] :
( ( image_set_a_set_a @ F @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% image_insert
thf(fact_821_finite__Int,axiom,
! [F2: set_real,G2: set_real] :
( ( ( finite_finite_real @ F2 )
| ( finite_finite_real @ G2 ) )
=> ( finite_finite_real @ ( inf_inf_set_real @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_822_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_823_finite__Int,axiom,
! [F2: set_int,G2: set_int] :
( ( ( finite_finite_int @ F2 )
| ( finite_finite_int @ G2 ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_824_finite__Int,axiom,
! [F2: set_set_a,G2: set_set_a] :
( ( ( finite_finite_set_a @ F2 )
| ( finite_finite_set_a @ G2 ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_825_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_826_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_827_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_828_Int__insert__right__if1,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_829_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_830_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_831_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_832_Int__insert__right__if0,axiom,
! [A2: real,A: set_real,B: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( inf_inf_set_real @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_833_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_834_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_835_insert__inter__insert,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_836_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_837_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_838_Int__insert__left__if1,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( insert_real @ A2 @ ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_839_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_840_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_841_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_842_Int__insert__left__if0,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ~ ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_843_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_844_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_845_Pow__Int__eq,axiom,
! [A: set_a,B: set_a] :
( ( pow_a @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_846_card__sumset__singleton__eq,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) )
& ( ~ ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_847_disjoint__insert_I2_J,axiom,
! [A: set_real,B2: real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A @ ( insert_real @ B2 @ B ) ) )
= ( ~ ( member_real @ B2 @ A )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_848_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_849_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_850_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_851_disjoint__insert_I2_J,axiom,
! [A: set_int,B2: int,B: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ A @ ( insert_int @ B2 @ B ) ) )
= ( ~ ( member_int @ B2 @ A )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_852_disjoint__insert_I1_J,axiom,
! [B: set_real,A2: real,A: set_real] :
( ( ( inf_inf_set_real @ B @ ( insert_real @ A2 @ A ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B )
& ( ( inf_inf_set_real @ B @ A )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_853_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_854_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_855_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_856_disjoint__insert_I1_J,axiom,
! [B: set_int,A2: int,A: set_int] :
( ( ( inf_inf_set_int @ B @ ( insert_int @ A2 @ A ) )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B )
& ( ( inf_inf_set_int @ B @ A )
= bot_bot_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_857_insert__disjoint_I2_J,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B ) )
= ( ~ ( member_real @ A2 @ B )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_858_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_859_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_860_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_861_insert__disjoint_I2_J,axiom,
! [A2: int,A: set_int,B: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B ) )
= ( ~ ( member_int @ A2 @ B )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_862_insert__disjoint_I1_J,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B )
& ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_863_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_864_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_865_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_866_insert__disjoint_I1_J,axiom,
! [A2: int,A: set_int,B: set_int] :
( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B )
& ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_867_card_Oempty,axiom,
( ( finite_card_real @ bot_bot_set_real )
= zero_zero_nat ) ).
% card.empty
thf(fact_868_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_869_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_870_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_871_card_Oempty,axiom,
( ( finite_card_int @ bot_bot_set_int )
= zero_zero_nat ) ).
% card.empty
thf(fact_872_card_Oinfinite,axiom,
! [A: set_real] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_card_real @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_873_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_874_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_875_card_Oinfinite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_card_int @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_876_card_Oinfinite,axiom,
! [A: set_set_a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_card_set_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_877_Diff__disjoint,axiom,
! [A: set_real,B: set_real] :
( ( inf_inf_set_real @ A @ ( minus_minus_set_real @ B @ A ) )
= bot_bot_set_real ) ).
% Diff_disjoint
thf(fact_878_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_879_Diff__disjoint,axiom,
! [A: set_int,B: set_int] :
( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B @ A ) )
= bot_bot_set_int ) ).
% Diff_disjoint
thf(fact_880_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_881_Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_882_card__0__eq,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( ( finite_card_real @ A )
= zero_zero_nat )
= ( A = bot_bot_set_real ) ) ) ).
% card_0_eq
thf(fact_883_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_884_card__0__eq,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( finite_card_set_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_885_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_886_card__0__eq,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( A = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_887_sumset__Int__carrier,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier
thf(fact_888_sumset__Int__carrier__eq_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_889_sumset__Int__carrier__eq_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_890_sumset__is__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_891_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_892_IntE,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A @ B ) )
=> ~ ( ( member_real @ C @ A )
=> ~ ( member_real @ C @ B ) ) ) ).
% IntE
thf(fact_893_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_894_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_895_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_896_IntD1,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C @ A ) ) ).
% IntD1
thf(fact_897_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_898_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_899_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_900_IntD2,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C @ B ) ) ).
% IntD2
thf(fact_901_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_902_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_903_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_904_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_905_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A6 ) ) ) ).
% Int_commute
thf(fact_906_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_907_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_908_image__Int__subset,axiom,
! [F: set_a > real,A: set_set_a,B: set_set_a] : ( ord_less_eq_set_real @ ( image_set_a_real @ F @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_real @ ( image_set_a_real @ F @ A ) @ ( image_set_a_real @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_909_image__Int__subset,axiom,
! [F: a > set_a,A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_910_image__Int__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_911_image__Pow__surj,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ( image_a_a @ F @ A )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_912_image__Pow__surj,axiom,
! [F: set_a > real,A: set_set_a,B: set_real] :
( ( ( image_set_a_real @ F @ A )
= B )
=> ( ( image_3546087905283185883t_real @ ( image_set_a_real @ F ) @ ( pow_set_a @ A ) )
= ( pow_real @ B ) ) ) ).
% image_Pow_surj
thf(fact_913_image__Pow__surj,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ( image_a_set_a @ F @ A )
= B )
=> ( ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A ) )
= ( pow_set_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_914_Int__emptyI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ~ ( member_real @ X2 @ B ) )
=> ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_915_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_916_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_917_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_918_Int__emptyI,axiom,
! [A: set_int,B: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ~ ( member_int @ X2 @ B ) )
=> ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int ) ) ).
% Int_emptyI
thf(fact_919_disjoint__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X: real] :
( ( member_real @ X @ A )
=> ~ ( member_real @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_920_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_921_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_922_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_923_disjoint__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int )
= ( ! [X: int] :
( ( member_int @ X @ A )
=> ~ ( member_int @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_924_Int__empty__left,axiom,
! [B: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_925_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_926_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_927_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_928_Int__empty__left,axiom,
! [B: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ B )
= bot_bot_set_int ) ).
% Int_empty_left
thf(fact_929_Int__empty__right,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ bot_bot_set_real )
= bot_bot_set_real ) ).
% Int_empty_right
thf(fact_930_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_931_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_932_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_933_Int__empty__right,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ A @ bot_bot_set_int )
= bot_bot_set_int ) ).
% Int_empty_right
thf(fact_934_disjoint__iff__not__equal,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X: real] :
( ( member_real @ X @ A )
=> ! [Y3: real] :
( ( member_real @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_935_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_936_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_937_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_938_disjoint__iff__not__equal,axiom,
! [A: set_int,B: set_int] :
( ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int )
= ( ! [X: int] :
( ( member_int @ X @ A )
=> ! [Y3: int] :
( ( member_int @ Y3 @ B )
=> ( X != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_939_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_940_Int__Collect__mono,axiom,
! [A: set_int,B: set_int,P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B @ ( collect_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_941_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_942_Int__Collect__mono,axiom,
! [A: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_943_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_944_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_945_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_946_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_947_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_948_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_949_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_950_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_951_Int__insert__right,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B ) ) ) )
& ( ~ ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_952_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_953_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_954_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_955_Int__insert__left,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ( ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( insert_real @ A2 @ ( inf_inf_set_real @ B @ C2 ) ) ) )
& ( ~ ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_956_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_957_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_958_Int__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_959_Int__Diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_960_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_961_Diff__Int2,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A @ C2 ) @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_962_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_963_Diff__Diff__Int,axiom,
! [A: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_964_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_965_Diff__Int__distrib,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ C2 @ A ) @ ( inf_inf_set_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_966_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_967_Diff__Int__distrib2,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ C2 )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A @ C2 ) @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_968_Cantors__paradox,axiom,
! [A: set_a] :
~ ? [F4: a > set_a] :
( ( image_a_set_a @ F4 @ A )
= ( pow_a @ A ) ) ).
% Cantors_paradox
thf(fact_969_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_970_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: real,F: a > real] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_real @ B2 @ ( image_a_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_971_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_972_rev__image__eqI,axiom,
! [X3: real,A: set_real,B2: a,F: real > a] :
( ( member_real @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_real_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_973_rev__image__eqI,axiom,
! [X3: real,A: set_real,B2: real,F: real > real] :
( ( member_real @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_974_rev__image__eqI,axiom,
! [X3: real,A: set_real,B2: nat,F: real > nat] :
( ( member_real @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_real_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_975_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_976_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: real,F: nat > real] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_977_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_978_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: set_a,F: a > set_a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_979_ball__imageD,axiom,
! [F: set_a > real,A: set_set_a,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_set_a_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: set_a] :
( ( member_set_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_980_ball__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_981_image__cong,axiom,
! [M2: set_a,N: set_a,F: a > set_a,G3: a > set_a] :
( ( M2 = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) )
=> ( ( image_a_set_a @ F @ M2 )
= ( image_a_set_a @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_982_image__cong,axiom,
! [M2: set_set_a,N: set_set_a,F: set_a > real,G3: set_a > real] :
( ( M2 = N )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) )
=> ( ( image_set_a_real @ F @ M2 )
= ( image_set_a_real @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_983_bex__imageD,axiom,
! [F: set_a > real,A: set_set_a,P: real > $o] :
( ? [X5: real] :
( ( member_real @ X5 @ ( image_set_a_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_984_bex__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_a_set_a @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_985_image__iff,axiom,
! [Z: set_a,F: a > set_a,A: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_986_image__iff,axiom,
! [Z: real,F: set_a > real,A: set_set_a] :
( ( member_real @ Z @ ( image_set_a_real @ F @ A ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_987_imageI,axiom,
! [X3: a,A: set_a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_988_imageI,axiom,
! [X3: a,A: set_a,F: a > real] :
( ( member_a @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ ( image_a_real @ F @ A ) ) ) ).
% imageI
thf(fact_989_imageI,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_990_imageI,axiom,
! [X3: real,A: set_real,F: real > a] :
( ( member_real @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_real_a @ F @ A ) ) ) ).
% imageI
thf(fact_991_imageI,axiom,
! [X3: real,A: set_real,F: real > real] :
( ( member_real @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_992_imageI,axiom,
! [X3: real,A: set_real,F: real > nat] :
( ( member_real @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_real_nat @ F @ A ) ) ) ).
% imageI
thf(fact_993_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_994_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > real] :
( ( member_nat @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ ( image_nat_real @ F @ A ) ) ) ).
% imageI
thf(fact_995_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_996_imageI,axiom,
! [X3: a,A: set_a,F: a > set_a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_997_image__Pow__mono,axiom,
! [F: set_a > real,A: set_set_a,B: set_real] :
( ( ord_less_eq_set_real @ ( image_set_a_real @ F @ A ) @ B )
=> ( ord_le3558479182127378552t_real @ ( image_3546087905283185883t_real @ ( image_set_a_real @ F ) @ ( pow_set_a @ A ) ) @ ( pow_real @ B ) ) ) ).
% image_Pow_mono
thf(fact_998_image__Pow__mono,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_999_image__Pow__mono,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_1000_Diff__triv,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
=> ( ( minus_minus_set_real @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1001_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1002_Diff__triv,axiom,
! [A: set_int,B: set_int] :
( ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int )
=> ( ( minus_minus_set_int @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1003_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1004_Diff__triv,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
=> ( ( minus_5736297505244876581_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1005_Int__Diff__disjoint,axiom,
! [A: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ A @ B ) @ ( minus_minus_set_real @ A @ B ) )
= bot_bot_set_real ) ).
% Int_Diff_disjoint
thf(fact_1006_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1007_Int__Diff__disjoint,axiom,
! [A: set_int,B: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B ) @ ( minus_minus_set_int @ A @ B ) )
= bot_bot_set_int ) ).
% Int_Diff_disjoint
thf(fact_1008_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1009_Int__Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ B ) )
= bot_bot_set_set_a ) ).
% Int_Diff_disjoint
thf(fact_1010_in__image__insert__iff,axiom,
! [B: set_set_real,X3: real,A: set_real] :
( ! [C4: set_real] :
( ( member_set_real @ C4 @ B )
=> ~ ( member_real @ X3 @ C4 ) )
=> ( ( member_set_real @ A @ ( image_2436557299294012491t_real @ ( insert_real @ X3 ) @ B ) )
= ( ( member_real @ X3 @ A )
& ( member_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1011_in__image__insert__iff,axiom,
! [B: set_set_nat,X3: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X3 @ C4 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B ) )
= ( ( member_nat @ X3 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1012_in__image__insert__iff,axiom,
! [B: set_set_int,X3: int,A: set_int] :
( ! [C4: set_int] :
( ( member_set_int @ C4 @ B )
=> ~ ( member_int @ X3 @ C4 ) )
=> ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X3 ) @ B ) )
= ( ( member_int @ X3 @ A )
& ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1013_in__image__insert__iff,axiom,
! [B: set_set_a,X3: a,A: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X3 @ C4 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ B ) )
= ( ( member_a @ X3 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1014_in__image__insert__iff,axiom,
! [B: set_set_set_a,X3: set_a,A: set_set_a] :
( ! [C4: set_set_a] :
( ( member_set_set_a @ C4 @ B )
=> ~ ( member_set_a @ X3 @ C4 ) )
=> ( ( member_set_set_a @ A @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X3 ) @ B ) )
= ( ( member_set_a @ X3 @ A )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1015_card__eq__0__iff,axiom,
! [A: set_real] :
( ( ( finite_card_real @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_real )
| ~ ( finite_finite_real @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1016_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1017_card__eq__0__iff,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1018_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1019_card__eq__0__iff,axiom,
! [A: set_int] :
( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_int )
| ~ ( finite_finite_int @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1020_all__subset__image,axiom,
! [F: set_a > real,A: set_set_a,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_set_a_real @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A )
=> ( P @ ( image_set_a_real @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1021_all__subset__image,axiom,
! [F: a > set_a,A: set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1022_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1023_subset__image__iff,axiom,
! [B: set_real,F: set_a > real,A: set_set_a] :
( ( ord_less_eq_set_real @ B @ ( image_set_a_real @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_set_a_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1024_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1025_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1026_image__subset__iff,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1027_image__subset__iff,axiom,
! [F: set_a > real,A: set_set_a,B: set_real] :
( ( ord_less_eq_set_real @ ( image_set_a_real @ F @ A ) @ B )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_real @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1028_subset__imageE,axiom,
! [B: set_real,F: set_a > real,A: set_set_a] :
( ( ord_less_eq_set_real @ B @ ( image_set_a_real @ F @ A ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
=> ( B
!= ( image_set_a_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1029_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1030_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1031_image__subsetI,axiom,
! [A: set_a,F: a > real,B: set_real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1032_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1033_image__subsetI,axiom,
! [A: set_real,F: real > real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1034_image__subsetI,axiom,
! [A: set_real,F: real > nat,B: set_nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1035_image__subsetI,axiom,
! [A: set_nat,F: nat > real,B: set_real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1036_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1037_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1038_image__subsetI,axiom,
! [A: set_real,F: real > a,B: set_a] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1039_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1040_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1041_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > real] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_real @ ( image_set_a_real @ F @ A ) @ ( image_set_a_real @ F @ B ) ) ) ).
% image_mono
thf(fact_1042_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1043_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1044_finite__surj,axiom,
! [A: set_real,B: set_real,F: real > real] :
( ( finite_finite_real @ A )
=> ( ( ord_less_eq_set_real @ B @ ( image_real_real @ F @ A ) )
=> ( finite_finite_real @ B ) ) ) ).
% finite_surj
thf(fact_1045_finite__surj,axiom,
! [A: set_real,B: set_nat,F: real > nat] :
( ( finite_finite_real @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_real_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1046_finite__surj,axiom,
! [A: set_real,B: set_int,F: real > int] :
( ( finite_finite_real @ A )
=> ( ( ord_less_eq_set_int @ B @ ( image_real_int @ F @ A ) )
=> ( finite_finite_int @ B ) ) ) ).
% finite_surj
thf(fact_1047_finite__surj,axiom,
! [A: set_a,B: set_real,F: a > real] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_real @ B @ ( image_a_real @ F @ A ) )
=> ( finite_finite_real @ B ) ) ) ).
% finite_surj
thf(fact_1048_finite__surj,axiom,
! [A: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1049_finite__surj,axiom,
! [A: set_a,B: set_int,F: a > int] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_int @ B @ ( image_a_int @ F @ A ) )
=> ( finite_finite_int @ B ) ) ) ).
% finite_surj
thf(fact_1050_finite__surj,axiom,
! [A: set_nat,B: set_real,F: nat > real] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_real @ B @ ( image_nat_real @ F @ A ) )
=> ( finite_finite_real @ B ) ) ) ).
% finite_surj
thf(fact_1051_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1052_finite__surj,axiom,
! [A: set_nat,B: set_int,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_int @ B @ ( image_nat_int @ F @ A ) )
=> ( finite_finite_int @ B ) ) ) ).
% finite_surj
thf(fact_1053_finite__surj,axiom,
! [A: set_int,B: set_real,F: int > real] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_real @ B @ ( image_int_real @ F @ A ) )
=> ( finite_finite_real @ B ) ) ) ).
% finite_surj
thf(fact_1054_finite__subset__image,axiom,
! [B: set_real,F: real > real,A: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ ( image_real_real @ F @ A ) )
=> ? [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A )
& ( finite_finite_real @ C4 )
& ( B
= ( image_real_real @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1055_finite__subset__image,axiom,
! [B: set_real,F: nat > real,A: set_nat] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ ( image_nat_real @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_real @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1056_finite__subset__image,axiom,
! [B: set_real,F: int > real,A: set_int] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ ( image_int_real @ F @ A ) )
=> ? [C4: set_int] :
( ( ord_less_eq_set_int @ C4 @ A )
& ( finite_finite_int @ C4 )
& ( B
= ( image_int_real @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1057_finite__subset__image,axiom,
! [B: set_nat,F: real > nat,A: set_real] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_real_nat @ F @ A ) )
=> ? [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A )
& ( finite_finite_real @ C4 )
& ( B
= ( image_real_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1058_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1059_finite__subset__image,axiom,
! [B: set_nat,F: int > nat,A: set_int] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_int_nat @ F @ A ) )
=> ? [C4: set_int] :
( ( ord_less_eq_set_int @ C4 @ A )
& ( finite_finite_int @ C4 )
& ( B
= ( image_int_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1060_finite__subset__image,axiom,
! [B: set_int,F: real > int,A: set_real] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ B @ ( image_real_int @ F @ A ) )
=> ? [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A )
& ( finite_finite_real @ C4 )
& ( B
= ( image_real_int @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1061_finite__subset__image,axiom,
! [B: set_int,F: nat > int,A: set_nat] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ B @ ( image_nat_int @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_int @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1062_finite__subset__image,axiom,
! [B: set_int,F: int > int,A: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ B @ ( image_int_int @ F @ A ) )
=> ? [C4: set_int] :
( ( ord_less_eq_set_int @ C4 @ A )
& ( finite_finite_int @ C4 )
& ( B
= ( image_int_int @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1063_finite__subset__image,axiom,
! [B: set_real,F: a > real,A: set_a] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ ( image_a_real @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_real @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1064_ex__finite__subset__image,axiom,
! [F: real > real,A: set_real,P: set_real > $o] :
( ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A )
& ( P @ ( image_real_real @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1065_ex__finite__subset__image,axiom,
! [F: nat > real,A: set_nat,P: set_real > $o] :
( ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1066_ex__finite__subset__image,axiom,
! [F: int > real,A: set_int,P: set_real > $o] :
( ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_int_real @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A )
& ( P @ ( image_int_real @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1067_ex__finite__subset__image,axiom,
! [F: real > nat,A: set_real,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_real_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A )
& ( P @ ( image_real_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1068_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1069_ex__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A )
& ( P @ ( image_int_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1070_ex__finite__subset__image,axiom,
! [F: real > int,A: set_real,P: set_int > $o] :
( ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_real_int @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A )
& ( P @ ( image_real_int @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1071_ex__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_int @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1072_ex__finite__subset__image,axiom,
! [F: int > int,A: set_int,P: set_int > $o] :
( ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_int] :
( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A )
& ( P @ ( image_int_int @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1073_ex__finite__subset__image,axiom,
! [F: a > real,A: set_a,P: set_real > $o] :
( ( ? [B5: set_real] :
( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_a_real @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_real @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1074_all__finite__subset__image,axiom,
! [F: real > real,A: set_real,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A ) )
=> ( P @ ( image_real_real @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1075_all__finite__subset__image,axiom,
! [F: nat > real,A: set_nat,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1076_all__finite__subset__image,axiom,
! [F: int > real,A: set_int,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_int_real @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A ) )
=> ( P @ ( image_int_real @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1077_all__finite__subset__image,axiom,
! [F: real > nat,A: set_real,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_real_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A ) )
=> ( P @ ( image_real_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1078_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1079_all__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A ) )
=> ( P @ ( image_int_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1080_all__finite__subset__image,axiom,
! [F: real > int,A: set_real,P: set_int > $o] :
( ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_real_int @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ A ) )
=> ( P @ ( image_real_int @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1081_all__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_int @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1082_all__finite__subset__image,axiom,
! [F: int > int,A: set_int,P: set_int > $o] :
( ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_int] :
( ( ( finite_finite_int @ B5 )
& ( ord_less_eq_set_int @ B5 @ A ) )
=> ( P @ ( image_int_int @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1083_all__finite__subset__image,axiom,
! [F: a > real,A: set_a,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ( finite_finite_real @ B5 )
& ( ord_less_eq_set_real @ B5 @ ( image_a_real @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_real @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1084_image__diff__subset,axiom,
! [F: set_a > real,A: set_set_a,B: set_set_a] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ ( image_set_a_real @ F @ A ) @ ( image_set_a_real @ F @ B ) ) @ ( image_set_a_real @ F @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1085_image__diff__subset,axiom,
! [F: a > set_a,A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1086_image__diff__subset,axiom,
! [F: set_a > set_a,A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) @ ( image_set_a_set_a @ F @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1087_image__diff__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1088_image__diff__subset,axiom,
! [F: set_a > a,A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) @ ( image_set_a_a @ F @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1089_card__image__le,axiom,
! [A: set_real,F: real > a] :
( ( finite_finite_real @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_real_a @ F @ A ) ) @ ( finite_card_real @ A ) ) ) ).
% card_image_le
thf(fact_1090_card__image__le,axiom,
! [A: set_a,F: a > set_a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( image_a_set_a @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_1091_card__image__le,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_1092_card__image__le,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_1093_card__image__le,axiom,
! [A: set_int,F: int > a] :
( ( finite_finite_int @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_int_a @ F @ A ) ) @ ( finite_card_int @ A ) ) ) ).
% card_image_le
thf(fact_1094_card__image__le,axiom,
! [A: set_set_a,F: set_a > real] :
( ( finite_finite_set_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_real @ ( image_set_a_real @ F @ A ) ) @ ( finite_card_set_a @ A ) ) ) ).
% card_image_le
thf(fact_1095_card__image__le,axiom,
! [A: set_set_a,F: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_set_a_a @ F @ A ) ) @ ( finite_card_set_a @ A ) ) ) ).
% card_image_le
thf(fact_1096_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1097_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1098_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1099_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1100_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1101_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1102_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1103_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1104_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1105_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1106_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1107_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1108_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1109_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_1110_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1111_diff__ge__0__iff__ge,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1112_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1113_KS__def,axiom,
( ks
= ( image_set_a_real
@ ^ [A6: set_a] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A6 ) ) )
@ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ) ).
% KS_def
thf(fact_1114_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1115_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1116_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1117_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1118_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1119_finite__Collect__conjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1120_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1121_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1122_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1123_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1124_finite__Collect__disjI,axiom,
! [P: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1125_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1126_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1127_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1128_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1129_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1130_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X: a] : ( member_a @ X @ A6 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ) ).
% sumset_def
thf(fact_1131_sumsetp__sumset__eq,axiom,
! [A: set_a,B: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X: a] : ( member_a @ X @ A )
@ ^ [X: a] : ( member_a @ X @ B ) )
= ( ^ [X: a] : ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_1132_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1133_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_1134_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_1135_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_1136_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_1137_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1138_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_1139_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_1140_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_1141_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1142_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1143_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1144_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1145_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1146_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1147_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1148_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1149_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1150_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1151_singleton__conv,axiom,
! [A2: real] :
( ( collect_real
@ ^ [X: real] : ( X = A2 ) )
= ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singleton_conv
thf(fact_1152_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X: a] : ( X = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1153_singleton__conv,axiom,
! [A2: set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( X = A2 ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_1154_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1155_singleton__conv,axiom,
! [A2: int] :
( ( collect_int
@ ^ [X: int] : ( X = A2 ) )
= ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singleton_conv
thf(fact_1156_singleton__conv2,axiom,
! [A2: real] :
( ( collect_real
@ ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singleton_conv2
thf(fact_1157_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_1158_singleton__conv2,axiom,
! [A2: set_a] :
( ( collect_set_a
@ ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_1159_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1160_singleton__conv2,axiom,
! [A2: int] :
( ( collect_int
@ ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singleton_conv2
thf(fact_1161_finite__Collect__subsets,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( finite9007344921179782393t_real
@ ( collect_set_real
@ ^ [B5: set_real] : ( ord_less_eq_set_real @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1162_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1163_finite__Collect__subsets,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1164_finite__Collect__subsets,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B5: set_set_a] : ( ord_le3724670747650509150_set_a @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1165_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B5: set_a] : ( ord_less_eq_set_a @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1166_Min__const,axiom,
! [A: set_real,C: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( lattic3629708407755379051n_real
@ ( image_real_real
@ ^ [Uu: real] : C
@ A ) )
= C ) ) ) ).
% Min_const
thf(fact_1167_Min__const,axiom,
! [A: set_a,C: real] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( lattic3629708407755379051n_real
@ ( image_a_real
@ ^ [Uu: a] : C
@ A ) )
= C ) ) ) ).
% Min_const
thf(fact_1168_Min__const,axiom,
! [A: set_set_a,C: real] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic3629708407755379051n_real
@ ( image_set_a_real
@ ^ [Uu: set_a] : C
@ A ) )
= C ) ) ) ).
% Min_const
thf(fact_1169_Min__const,axiom,
! [A: set_nat,C: real] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic3629708407755379051n_real
@ ( image_nat_real
@ ^ [Uu: nat] : C
@ A ) )
= C ) ) ) ).
% Min_const
thf(fact_1170_Min__const,axiom,
! [A: set_int,C: real] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic3629708407755379051n_real
@ ( image_int_real
@ ^ [Uu: int] : C
@ A ) )
= C ) ) ) ).
% Min_const
thf(fact_1171_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1172_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1173_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1174_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1175_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1176_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1177_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1178_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ A6 )
@ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1179_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A6 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1180_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A6 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1181_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1182_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1183_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1184_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1185_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1186_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1187_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_1188_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_1189_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1190_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1191_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1192_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1193_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1194_A__def,axiom,
( a2
= ( fChoice_set_a
@ ^ [A6: set_a] :
( ( member_set_a @ A6 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) )
& ( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A6 ) ) ) ) ) ) ) ).
% A_def
thf(fact_1195_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1196_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1197_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1198_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1199_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1200_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1201_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1202_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1203_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1204_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1205_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1206_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1207_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1208_sumset,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A5: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B4: a] : ( insert_a @ ( addition @ A5 @ B4 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ g ) ) )
@ ( inf_inf_set_a @ A @ g ) ) ) ) ).
% sumset
thf(fact_1209_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_1210_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1211_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_1212_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1213_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1214_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1215_zdiv__int,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zdiv_int
thf(fact_1216_real__divide__square__eq,axiom,
! [R: real,A2: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A2 ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A2 @ R ) ) ).
% real_divide_square_eq
thf(fact_1217_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1218_real__of__nat__div2,axiom,
! [N2: nat,X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X3 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1219_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z3: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ? [Y4: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y4 ) )
& ! [Z3: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_1220_real__of__nat__div4,axiom,
! [N2: nat,X3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X3 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% real_of_nat_div4
thf(fact_1221_sumset__subset__Un_I1_J,axiom,
! [A: set_a,B: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% sumset_subset_Un(1)
thf(fact_1222_sumset__subset__Un_I2_J,axiom,
! [A: set_a,B: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ C2 ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_1223_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_1224_sumset__subset__Un1,axiom,
! [A: set_a,A7: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ A7 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_1225_sumset__subset__Un2,axiom,
! [A: set_a,B: set_a,B7: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ B7 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B7 ) ) ) ).
% sumset_subset_Un2
thf(fact_1226_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1227_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1228_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1229_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_1230_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N4: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1231_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1232_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1233_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1234_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_1235_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1236_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1237_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1238_sumset__eq,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= ( collect_a
@ ^ [C5: a] :
? [X: a] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ g ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( inf_inf_set_a @ B @ g ) )
& ( C5
= ( addition @ X @ Y3 ) ) ) ) ) ) ).
% sumset_eq
thf(fact_1239_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1240_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_1241_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1242_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1243_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1244_int__ops_I8_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(8)
thf(fact_1245_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1246_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1247_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1248_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1249_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1250_conj__le__cong,axiom,
! [X3: int,X6: int,P: $o,P2: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1251_imp__le__cong,axiom,
! [X3: int,X6: int,P: $o,P2: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1252_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_1253_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_1254_Ruzsa__triangle__ineq2,axiom,
! [U2: set_a,V3: set_a,W2: set_a] :
( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ g )
=> ( ( U2 != bot_bot_set_a )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ g )
=> ( ( finite_finite_a @ W2 )
=> ( ( ord_less_eq_set_a @ W2 @ g )
=> ( ord_less_eq_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ W2 ) @ ( times_times_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ U2 ) @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ U2 @ W2 ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq2
thf(fact_1255_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_1256_left__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ zero @ A2 )
= A2 ) ) ).
% left_unit
thf(fact_1257_right__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ A2 @ zero )
= A2 ) ) ).
% right_unit
thf(fact_1258_sumset__D_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(1)
thf(fact_1259_sumset__D_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(2)
thf(fact_1260_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_1261_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_1262_Ruzsa__triangle__ineq1,axiom,
! [U2: set_a,V3: set_a,W2: set_a] :
( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ g )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ g )
=> ( ( finite_finite_a @ W2 )
=> ( ( ord_less_eq_set_a @ W2 @ g )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ U2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ V3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W2 ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ V3 ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W2 ) ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq1
thf(fact_1263_minusset__distrib__sum,axiom,
! [A: set_a,B: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% minusset_distrib_sum
thf(fact_1264_finite__minusset,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ).
% finite_minusset
thf(fact_1265_minusset__subset__carrier,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ g ) ).
% minusset_subset_carrier
thf(fact_1266_card__differenceset__commute,axiom,
! [B: set_a,A: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).
% card_differenceset_commute
thf(fact_1267_finite__differenceset,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).
% finite_differenceset
thf(fact_1268_card__minusset_H,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_minusset'
thf(fact_1269_differenceset__commute,axiom,
! [B: set_a,A: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% differenceset_commute
thf(fact_1270_minus__minusset,axiom,
! [A: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( inf_inf_set_a @ A @ g ) ) ).
% minus_minusset
thf(fact_1271_minusset__is__empty__iff,axiom,
! [A: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_1272_card__minusset,axiom,
! [A: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) ).
% card_minusset
thf(fact_1273_minusset__triv,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% minusset_triv
% Helper facts (4)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: set_a > $o] :
( ( P @ ( fChoice_set_a @ P ) )
= ( ? [X7: set_a] : ( P @ X7 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,conjecture,
thesisa ).
%------------------------------------------------------------------------------