TPTP Problem File: SLH0779^1.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_00069_002251__12007740_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1469 ( 553 unt; 183 typ; 0 def)
% Number of atoms : 3799 (1270 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 12607 ( 371 ~; 47 |; 320 &;10126 @)
% ( 0 <=>;1743 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 1197 (1197 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 14 con; 0-7 aty)
% Number of variables : 3943 ( 512 ^;3318 !; 113 ?;3943 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:17:08.401
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $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_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
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_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_Itf__a_J,type,
condit3373647341569784514_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
finite_Fpow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite312795530508511377_set_a: ( a > set_set_a > set_set_a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_001tf__a_001t__Set__Oset_Itf__a_J,type,
finite3518785373051244337_set_a: ( a > set_a > set_a ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $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_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_Finite__Set_Ofold_001t__Nat__Onat_001_Eo,type,
finite_fold_nat_o: ( nat > $o > $o ) > $o > set_nat > $o ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite4178521680790401110et_nat: ( nat > set_set_nat > set_set_nat ) > set_set_nat > set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
finite4864421574810880708_set_a: ( nat > set_a > set_a ) > set_a > set_nat > set_a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001tf__a,type,
finite_fold_nat_a: ( nat > a > a ) > a > set_nat > a ).
thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
finite5985231929012247624_set_a: ( set_a > set_a > set_a ) > set_a > set_set_a > set_a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite9006272623207878408_set_a: ( a > set_set_a > set_set_a ) > set_set_a > set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Set__Oset_Itf__a_J,type,
finite_fold_a_set_a: ( a > set_a > set_a ) > set_a > set_a > set_a ).
thf(sy_c_Finite__Set_Ofold__graph_001t__Nat__Onat_001tf__a,type,
finite9142365241556460134_nat_a: ( nat > a > a ) > a > set_nat > a > $o ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_001t__Nat__Onat,type,
comp_s4757170965337636338_a_nat: ( set_a > set_a > set_a ) > ( nat > set_a ) > nat > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_001tf__a,type,
comp_s1103547301056249180et_a_a: ( set_a > set_a > set_a ) > ( a > set_a ) > a > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001tf__a,type,
comp_set_a_set_a_a: ( set_a > set_a ) > ( a > set_a ) > a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001tf__a_001tf__a,type,
comp_set_a_a_a: ( set_a > a ) > ( a > set_a ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_Itf__a_J_001tf__a,type,
comp_a_set_a_a: ( a > set_a ) > ( a > a ) > a > set_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Set__Oset_Itf__a_J,type,
fun_upd_a_set_a: ( a > set_a ) > a > set_a > a > set_a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
fun_upd_a_a: ( a > a ) > a > a > a > a ).
thf(sy_c_FuncSet_Oextensional_001tf__a_001tf__a,type,
extensional_a_a: set_a > set_a_a ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001t__Set__Oset_Itf__a_J,type,
restrict_a_set_a: ( a > set_a ) > set_a > a > set_a ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001tf__a,type,
restrict_a_a: ( a > a ) > set_a > a > a ).
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_Group__Theory_Ocommutative__monoid__axioms_001tf__a,type,
group_2081300317213596122ioms_a: set_a > ( a > a > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup__axioms_001tf__a,type,
group_group_axioms_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001t__Nat__Onat,type,
group_Units_nat: set_nat > ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Omonoid__isomorphism_001tf__a_001t__Set__Oset_Itf__a_J,type,
group_2347938599424458858_set_a: ( a > set_a ) > set_a > ( a > a > a ) > a > set_set_a > ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_Group__Theory_Omonoid__isomorphism_001tf__a_001tf__a,type,
group_9128011940810589962sm_a_a: ( a > a ) > set_a > ( a > a > a ) > a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Set__Oset_Itf__a_J,type,
group_subgroup_set_a: set_set_a > set_set_a > ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Osubgroup__of__abelian__group_001tf__a,type,
group_2277603475229492062roup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
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__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_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
nO_MATCH_set_a_set_a: set_a > set_a > $o ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_OUniq_001tf__a,type,
uniq_a: ( a > $o ) > $o ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
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__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
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_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_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_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
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_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_I_062_Itf__a_Mtf__a_J_J,type,
bot_bot_set_a_a: set_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_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_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $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_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Partial__Function_Oflat__lub_001tf__a,type,
partial_flat_lub_a: a > set_a > a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Nat__Onat,type,
pluenn2073725187428264546up_nat: set_nat > ( nat > nat > nat ) > nat > $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_Osumset_001t__Nat__Onat,type,
pluenn3669378163024332905et_nat: set_nat > ( nat > nat > nat ) > set_nat > set_nat > set_nat ).
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_001t__Nat__Onat,type,
pluenn5670965976768739049tp_nat: set_nat > ( nat > nat > nat ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
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_Set_OBex_001tf__a,type,
bex_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
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_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__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
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_Obind_001tf__a_001t__Set__Oset_Itf__a_J,type,
bind_a_set_a: set_a > ( a > set_set_a ) > set_set_a ).
thf(sy_c_Set_Obind_001tf__a_001tf__a,type,
bind_a_a: set_a > ( a > set_a ) > set_a ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001tf__a,type,
filter_a: ( a > $o ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
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__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
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__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
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_Itf__a_J_001_062_Itf__a_M_Eo_J,type,
image_set_a_a_o: ( set_a > a > $o ) > set_set_a > set_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_Eo,type,
image_set_a_o: ( set_a > $o ) > set_set_a > set_o ).
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_001_062_Itf__a_M_Eo_J,type,
image_a_a_o: ( a > a > $o ) > set_a > set_a_o ).
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__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
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_001_062_Itf__a_Mtf__a_J,type,
insert_a_a: ( a > a ) > set_a_a > set_a_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_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_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set__Theory_Opartition_001tf__a,type,
set_partition_a: set_a > set_set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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__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_A,type,
a2: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_B_H,type,
b2: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1278)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_2_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_3_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_4_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_5_associative,axiom,
! [A: a,B4: a,C: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C )
= ( addition @ A @ ( addition @ B4 @ C ) ) ) ) ) ) ).
% associative
thf(fact_6_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_7_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_8_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_9_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_10_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_11_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_12_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_13_UnCI,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_14_Un__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_15_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_16_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_17_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_18_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_19_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_20_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_21_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_22_sup_Obounded__iff,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_23_sup_Obounded__iff,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_24_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_25_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_26_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_27_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_28_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_29_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_30_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_31_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_32_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_33_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_34_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_35_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_36_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_37_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_38_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_39_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_40_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_41_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_42_sup_OcoboundedI2,axiom,
! [C: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B4 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_43_sup_OcoboundedI2,axiom,
! [C: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_44_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_45_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_46_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_47_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_48_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_49_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_50_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_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_55_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_56_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_57_sup_Ocobounded2,axiom,
! [B4: nat,A: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_58_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_59_sup_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_60_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_61_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_62_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_63_sup_OboundedI,axiom,
! [B4: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_64_sup_OboundedE,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_65_sup_OboundedE,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B4 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_66_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_67_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_68_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_69_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_70_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_71_sup_Oabsorb2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_72_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_73_sup_Oabsorb1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_74_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_75_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_76_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_77_sup_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( sup_sup_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% sup.orderI
thf(fact_78_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_79_sup_OorderE,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( A
= ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_80_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_81_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_82_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_83_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_84_sup__mono,axiom,
! [A: set_a,C: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_85_sup__mono,axiom,
! [A: nat,C: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_86_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_87_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B4: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_88_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_89_le__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_90_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_91_le__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_92_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_93_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_94_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_95_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_96_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_97_le__supI,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_98_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_99_le__supE,axiom,
! [A: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_100_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_101_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_102_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_103_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_104_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_105_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_106_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_107_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_108_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_109_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_110_Un__least,axiom,
! [A2: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_111_Un__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_112_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_113_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_114_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_115_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_116_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_117_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_118_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_119_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_120_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_121_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_122_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_123_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_124_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_125_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_126_Un__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_127_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_128_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_129_UnI2,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_130_UnI1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_131_UnE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_132_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_133_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_134_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_135_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_136_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_137_sumsetp__sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_138_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sumset_def
thf(fact_139_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_140_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_141_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_142_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_143_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_144_nle__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_145_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_146_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_147_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_148_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_149_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_150_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_151_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_152_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_153_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_154_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_155_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_156_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_157_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_158_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_159_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_160_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_161_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_162_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_163_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_164_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_165_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_166_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_167_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_168_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_169_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_170_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_171_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_172_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_173_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_174_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_175_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_176_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_177_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_178_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) ) ) ).
% inf.assoc
thf(fact_179_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_180_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_181_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_182_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) ) ) ).
% inf.left_commute
thf(fact_183_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_184_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_185_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_186_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_187_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_188_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_189_Int__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_190_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_191_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_192_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_193_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_194_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_195_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_196_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_197_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_198_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_199_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_200_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_201_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_202_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_203_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_204_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_205_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_206_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_207_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
| ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_208_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_209_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_210_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_211_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_212_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_213_inf_OcoboundedI2,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_214_inf_OcoboundedI2,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_215_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_216_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_217_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_218_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_219_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_220_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_221_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_222_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_223_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_224_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_225_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_226_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_227_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_228_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_229_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_230_inf_OboundedI,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_231_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_232_inf_OboundedE,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_233_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_234_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_235_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_236_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_237_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_238_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_239_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_240_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_241_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_242_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_243_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_244_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_245_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_246_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_247_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_248_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_249_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_250_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_251_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_252_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_253_inf__mono,axiom,
! [A: set_a,C: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_254_inf__mono,axiom,
! [A: nat,C: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_255_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_256_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_257_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_258_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_259_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_260_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_261_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_262_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_263_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_264_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_265_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_266_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_267_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_268_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_269_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_270_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_271_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_272_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_273_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_274_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_275_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_276_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_277_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_278_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_279_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_280_Int__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_281_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_282_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_283_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_284_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_285_Un__Int__distrib2,axiom,
! [B: set_a,C2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_286_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_287_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_288_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_289_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_290_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_291_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_292_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_293_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_294_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_295_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_296_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_297_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_298_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_301_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_302_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_303_verit__la__disequality,axiom,
! [A: nat,B4: nat] :
( ( A = B4 )
| ~ ( ord_less_eq_nat @ A @ B4 )
| ~ ( ord_less_eq_nat @ B4 @ A ) ) ).
% verit_la_disequality
thf(fact_304_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_305_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_306_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_307_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_308_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_309_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_310_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_311_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_312_order__subst1,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_313_order__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_314_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_315_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_316_antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_317_antisym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_318_dual__order_Otrans,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C @ B4 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_319_dual__order_Otrans,axiom,
! [B4: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_320_dual__order_Oantisym,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_321_dual__order_Oantisym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_322_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_323_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_324_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_325_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_326_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_327_order_Otrans,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_328_order_Otrans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_329_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_330_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_331_ord__le__eq__trans,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_332_ord__le__eq__trans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_333_ord__eq__le__trans,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( A = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_334_ord__eq__le__trans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( A = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_335_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_336_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_337_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_338_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_339_sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( collect_a
@ ^ [C3: a] :
? [X3: a] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A2 @ g ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( inf_inf_set_a @ B @ g ) )
& ( C3
= ( addition @ X3 @ Y4 ) ) ) ) ) ) ).
% sumset_eq
thf(fact_340_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_341_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ g )
& ( ( addition @ U @ X3 )
= zero )
& ( ( addition @ X3 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_342_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_343_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_344_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_345_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_346_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_347_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_348_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_349_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_350_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_351_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_352_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_353_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_354_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_355_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_356_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_357_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_358_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_359_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_360_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_361_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_362_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_363_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_364_Bex__def,axiom,
( bex_a
= ( ^ [A6: set_a,P2: a > $o] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_365_additive__abelian__group_Osumset__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B )
= ( collect_nat
@ ^ [C3: nat] :
? [X3: nat] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A2 @ G ) )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ ( inf_inf_set_nat @ B @ G ) )
& ( C3
= ( Addition @ X3 @ Y4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset_eq
thf(fact_366_additive__abelian__group_Osumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( collect_a
@ ^ [C3: a] :
? [X3: a] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A2 @ G ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( inf_inf_set_a @ B @ G ) )
& ( C3
= ( Addition @ X3 @ Y4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset_eq
thf(fact_367_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_368_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_369_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_370_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_371_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_372_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_373_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_374_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_375_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_376_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_377_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_378_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_379_additive__abelian__group_Osumset__def,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition )
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( pluenn5670965976768739049tp_nat @ G @ Addition
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ) ).
% additive_abelian_group.sumset_def
thf(fact_380_additive__abelian__group_Osumset__def,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ) ).
% additive_abelian_group.sumset_def
thf(fact_381_additive__abelian__group_Osumsetp__sumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumsetp_sumset_eq
thf(fact_382_Units__def,axiom,
( ( group_Units_a @ g @ addition @ zero )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ g )
& ( group_invertible_a @ g @ addition @ zero @ U2 ) ) ) ) ).
% Units_def
thf(fact_383_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_384_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_385_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G2: a,H: a] :
( ( member_a @ G2 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G2 @ H ) @ G ) ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G2 ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G2 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_386_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_387_abelian__group_Ointro,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_4866109990395492029noid_a @ G @ Composition @ Unit )
=> ( group_201663378560352916roup_a @ G @ Composition @ Unit ) ) ) ).
% abelian_group.intro
thf(fact_388_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G3: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G3 @ Composition2 @ Unit2 )
& ( group_4866109990395492029noid_a @ G3 @ Composition2 @ Unit2 ) ) ) ) ).
% abelian_group_def
thf(fact_389_subgroup_Osubgroup__inverse__equality,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( member_a @ U @ G )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= ( group_inverse_a @ G @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_390_subgroup__transitive,axiom,
! [K2: set_a,H2: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_subgroup_a @ K2 @ H2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G @ Composition @ Unit )
=> ( group_subgroup_a @ K2 @ G @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_391_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_392_subgroup_Oaxioms_I2_J,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( group_group_a @ G @ Composition @ Unit ) ) ).
% subgroup.axioms(2)
thf(fact_393_subgroup_Osubgroup__inverse__iff,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ X ) @ G )
= ( member_a @ X @ G ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_394_monoid_Omem__UnitsI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_395_monoid_Omem__UnitsD,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
& ( member_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_396_monoid_Ogroup__of__Units,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_group_a @ ( group_Units_a @ M @ Composition @ Unit ) @ Composition @ Unit ) ) ).
% monoid.group_of_Units
thf(fact_397_monoid_OUnits__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_Units_nat @ M @ Composition @ Unit )
= ( collect_nat
@ ^ [U2: nat] :
( ( member_nat @ U2 @ M )
& ( group_invertible_nat @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_398_monoid_OUnits__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_Units_a @ M @ Composition @ Unit )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ M )
& ( group_invertible_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_399_monoid_OsubgroupI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G @ M )
=> ( ( member_a @ Unit @ G )
=> ( ! [G2: a,H: a] :
( ( member_a @ G2 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition @ G2 @ H ) @ G ) ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( group_invertible_a @ M @ Composition @ Unit @ G2 ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ G2 ) @ G ) )
=> ( group_subgroup_a @ G @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_400_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_401_Group__Theory_Omonoid__def,axiom,
( group_monoid_a
= ( ^ [M2: set_a,Composition2: a > a > a,Unit2: a] :
( ! [A4: a,B3: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B3 @ M2 )
=> ( member_a @ ( Composition2 @ A4 @ B3 ) @ M2 ) ) )
& ( member_a @ Unit2 @ M2 )
& ! [A4: a,B3: a,C3: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B3 @ M2 )
=> ( ( member_a @ C3 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A4 @ B3 ) @ C3 )
= ( Composition2 @ A4 @ ( Composition2 @ B3 @ C3 ) ) ) ) ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition2 @ Unit2 @ A4 )
= A4 ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition2 @ A4 @ Unit2 )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_402_monoid_Ocomposition__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B4: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B4 @ M )
=> ( member_a @ ( Composition @ A @ B4 ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_403_monoid_Oinverse__unique,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( member_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_404_monoid_Ounit__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( member_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_405_monoid_Oassociative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B4: a,C: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B4 @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B4 ) @ C )
= ( Composition @ A @ ( Composition @ B4 @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_406_monoid_Oright__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_407_monoid_Oleft__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_408_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ! [A3: a,B2: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B2 @ M )
=> ( member_a @ ( Composition @ A3 @ B2 ) @ M ) ) )
=> ( ( member_a @ Unit @ M )
=> ( ! [A3: a,B2: a,C4: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B2 @ M )
=> ( ( member_a @ C4 @ M )
=> ( ( Composition @ ( Composition @ A3 @ B2 ) @ C4 )
= ( Composition @ A3 @ ( Composition @ B2 @ C4 ) ) ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition @ Unit @ A3 )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition @ A3 @ Unit )
= A3 ) )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_409_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_410_commutative__monoid_Ocommutative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_411_monoid_Oinverse__equality,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_412_monoid_Oinverse__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ Unit )
= Unit ) ) ).
% monoid.inverse_unit
thf(fact_413_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_414_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_415_monoid_Ocomposition__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_416_monoid_Ounit__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_invertible_a @ M @ Composition @ Unit @ Unit ) ) ).
% monoid.unit_invertible
thf(fact_417_monoid_Oinvertible__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ M )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_418_monoid_OinvertibleI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_419_monoid_OinvertibleE,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_a @ V3 @ M ) )
=> ~ ( member_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_420_Group__Theory_Ogroup_Oaxioms_I1_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( group_monoid_a @ G @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(1)
thf(fact_421_group_Oinvertible,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( member_a @ U @ G )
=> ( group_invertible_a @ G @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_422_abelian__group_Oaxioms_I1_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G @ Composition @ Unit )
=> ( group_group_a @ G @ Composition @ Unit ) ) ).
% abelian_group.axioms(1)
thf(fact_423_commutative__monoid_Oaxioms_I1_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ).
% commutative_monoid.axioms(1)
thf(fact_424_abelian__group_Oaxioms_I2_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G @ Composition @ Unit )
=> ( group_4866109990395492029noid_a @ G @ Composition @ Unit ) ) ).
% abelian_group.axioms(2)
thf(fact_425_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_426_monoid_Oinverse__composition__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_427_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_428_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_429_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_430_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_431_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_432_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_433_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_434_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_435_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_436_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_437_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_438_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_439_image__eqI,axiom,
! [B4: set_a,F: a > set_a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_440_image__eqI,axiom,
! [B4: a,F: a > a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_441_image__ident,axiom,
! [Y5: set_a] :
( ( image_a_a
@ ^ [X3: a] : X3
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_442_if__image__distrib,axiom,
! [P: a > $o,F: a > set_a,G4: a > set_a,S: set_a] :
( ( image_a_set_a
@ ^ [X3: a] : ( if_set_a @ ( P @ X3 ) @ ( F @ X3 ) @ ( G4 @ X3 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_set_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_443_if__image__distrib,axiom,
! [P: nat > $o,F: nat > a,G4: nat > a,S: set_nat] :
( ( image_nat_a
@ ^ [X3: nat] : ( if_a @ ( P @ X3 ) @ ( F @ X3 ) @ ( G4 @ X3 ) )
@ S )
= ( sup_sup_set_a @ ( image_nat_a @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_a @ G4
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_444_if__image__distrib,axiom,
! [P: a > $o,F: a > a,G4: a > a,S: set_a] :
( ( image_a_a
@ ^ [X3: a] : ( if_a @ ( P @ X3 ) @ ( F @ X3 ) @ ( G4 @ X3 ) )
@ S )
= ( sup_sup_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_445_imageE,axiom,
! [B4: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B4
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_446_imageE,axiom,
! [B4: a,F: a > a,A2: set_a] :
( ( member_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B4
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_447_image__image,axiom,
! [F: set_a > a,G4: a > set_a,A2: set_a] :
( ( image_set_a_a @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X3: a] : ( F @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_448_image__image,axiom,
! [F: set_a > set_a,G4: a > set_a,A2: set_a] :
( ( image_set_a_set_a @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X3: a] : ( F @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_449_image__image,axiom,
! [F: a > a,G4: a > a,A2: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X3: a] : ( F @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_450_image__image,axiom,
! [F: a > set_a,G4: a > a,A2: set_a] :
( ( image_a_set_a @ F @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X3: a] : ( F @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_451_Compr__image__eq,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_set_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_452_Compr__image__eq,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_453_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_nat_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_454_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_a_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_455_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_456_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_457_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_458_ball__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_459_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_460_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G4: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_461_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_a,G4: a > set_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_462_bex__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_463_bex__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_464_image__iff,axiom,
! [Z: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_465_image__iff,axiom,
! [Z: a,F: a > a,A2: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_466_imageI,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_467_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_468_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_469_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_470_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_471_image__subsetI,axiom,
! [A2: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_472_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B
!= ( image_a_set_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_473_subset__imageE,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_474_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_475_image__subset__iff,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_476_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_477_subset__image__iff,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_478_image__Un,axiom,
! [F: a > set_a,A2: set_a,B: set_a] :
( ( image_a_set_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Un
thf(fact_479_image__Un,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_480_image__def,axiom,
( image_a_set_a
= ( ^ [F2: a > set_a,A6: set_a] :
( collect_set_a
@ ^ [Y4: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Y4
= ( F2 @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_481_image__def,axiom,
( image_a_a
= ( ^ [F2: a > a,A6: set_a] :
( collect_a
@ ^ [Y4: a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Y4
= ( F2 @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_482_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_483_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_484_subgroup_Oimage__of__inverse,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( member_a @ X @ G )
=> ( member_a @ X @ ( image_a_a @ ( group_inverse_a @ M @ Composition @ Unit ) @ G ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_485_monoid_Oinverse__undefined,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ~ ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= undefined_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_486_group_Oinverse__subgroupI,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G @ Composition @ Unit )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit ) ) ) ).
% group.inverse_subgroupI
thf(fact_487_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G @ Composition @ Unit ) )
=> ( group_subgroup_a @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_488_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_489_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B: set_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_490_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_491_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_492_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_493_commutative__monoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_2081300317213596122ioms_a @ M @ Composition )
=> ( group_4866109990395492029noid_a @ M @ Composition @ Unit ) ) ) ).
% commutative_monoid.intro
thf(fact_494_commutative__monoid__def,axiom,
( group_4866109990395492029noid_a
= ( ^ [M2: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M2 @ Composition2 @ Unit2 )
& ( group_2081300317213596122ioms_a @ M2 @ Composition2 ) ) ) ) ).
% commutative_monoid_def
thf(fact_495_Group__Theory_Ogroup_Ointro,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ G @ Composition @ Unit )
=> ( ( group_group_axioms_a @ G @ Composition @ Unit )
=> ( group_group_a @ G @ Composition @ Unit ) ) ) ).
% Group_Theory.group.intro
thf(fact_496_Group__Theory_Ogroup__def,axiom,
( group_group_a
= ( ^ [G3: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ G3 @ Composition2 @ Unit2 )
& ( group_group_axioms_a @ G3 @ Composition2 @ Unit2 ) ) ) ) ).
% Group_Theory.group_def
thf(fact_497_commutative__monoid__axioms_Ointro,axiom,
! [M: set_a,Composition: a > a > a] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ M )
=> ( ( member_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_2081300317213596122ioms_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_498_commutative__monoid__axioms__def,axiom,
( group_2081300317213596122ioms_a
= ( ^ [M2: set_a,Composition2: a > a > a] :
! [X3: a,Y4: a] :
( ( member_a @ X3 @ M2 )
=> ( ( member_a @ Y4 @ M2 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_499_Group__Theory_Ogroup__axioms__def,axiom,
( group_group_axioms_a
= ( ^ [G3: set_a,Composition2: a > a > a,Unit2: a] :
! [U2: a] :
( ( member_a @ U2 @ G3 )
=> ( group_invertible_a @ G3 @ Composition2 @ Unit2 @ U2 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_500_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ! [U3: a] :
( ( member_a @ U3 @ G )
=> ( group_invertible_a @ G @ Composition @ Unit @ U3 ) )
=> ( group_group_axioms_a @ G @ Composition @ Unit ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_501_Group__Theory_Ogroup_Oaxioms_I2_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( group_group_axioms_a @ G @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(2)
thf(fact_502_commutative__monoid_Oaxioms_I2_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_2081300317213596122ioms_a @ M @ Composition ) ) ).
% commutative_monoid.axioms(2)
thf(fact_503_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_504_prop__restrict,axiom,
! [X: a,Z4: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_505_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_506_Collect__restrict,axiom,
! [X5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_507_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_508_conj__subset__def,axiom,
! [A2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A2 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_509_subset__CollectI,axiom,
! [B: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_510_subset__CollectI,axiom,
! [B: set_a,A2: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_511_subset__Collect__iff,axiom,
! [B: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_512_subset__Collect__iff,axiom,
! [B: set_a,A2: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_513_Inf_OINF__identity__eq,axiom,
! [Inf: set_a > a,A2: set_a] :
( ( Inf
@ ( image_a_a
@ ^ [X3: a] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_514_Inf_OINF__cong,axiom,
! [A2: set_a,B: set_a,C2: a > a,D2: a > a,Inf: set_a > a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_a_a @ C2 @ A2 ) )
= ( Inf @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_515_Inf_OINF__cong,axiom,
! [A2: set_a,B: set_a,C2: a > set_a,D2: a > set_a,Inf: set_set_a > set_a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_a_set_a @ C2 @ A2 ) )
= ( Inf @ ( image_a_set_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_516_Sup_OSUP__cong,axiom,
! [A2: set_a,B: set_a,C2: a > a,D2: a > a,Sup: set_a > a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_a_a @ C2 @ A2 ) )
= ( Sup @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_517_Sup_OSUP__cong,axiom,
! [A2: set_a,B: set_a,C2: a > set_a,D2: a > set_a,Sup: set_set_a > set_a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_a_set_a @ C2 @ A2 ) )
= ( Sup @ ( image_a_set_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_518_Sup_OSUP__identity__eq,axiom,
! [Sup: set_a > a,A2: set_a] :
( ( Sup
@ ( image_a_a
@ ^ [X3: a] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_519_inverse__def,axiom,
( ( group_inverse_a @ g @ addition @ zero )
= ( restrict_a_a
@ ^ [U2: a] :
( the_a
@ ^ [V4: a] :
( ( member_a @ V4 @ g )
& ( ( addition @ U2 @ V4 )
= zero )
& ( ( addition @ V4 @ U2 )
= zero ) ) )
@ g ) ) ).
% inverse_def
thf(fact_520_subgroup__of__abelian__group_Oaxioms_I2_J,axiom,
! [H2: set_a,G: set_a,Composition: a > a > a,Unit: a] :
( ( group_2277603475229492062roup_a @ H2 @ G @ Composition @ Unit )
=> ( group_201663378560352916roup_a @ G @ Composition @ Unit ) ) ).
% subgroup_of_abelian_group.axioms(2)
thf(fact_521_image__Fpow__mono,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_522_image__Fpow__mono,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_523_monoid__isomorphism_Oimage__subgroup,axiom,
! [Eta: a > set_a,M: set_a,Composition: a > a > a,Unit: a,M3: set_set_a,Composition3: set_a > set_a > set_a,Unit3: set_a,G: set_a] :
( ( group_2347938599424458858_set_a @ Eta @ M @ Composition @ Unit @ M3 @ Composition3 @ Unit3 )
=> ( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( group_subgroup_set_a @ ( image_a_set_a @ Eta @ G ) @ M3 @ Composition3 @ Unit3 ) ) ) ).
% monoid_isomorphism.image_subgroup
thf(fact_524_monoid__isomorphism_Oimage__subgroup,axiom,
! [Eta: a > a,M: set_a,Composition: a > a > a,Unit: a,M3: set_a,Composition3: a > a > a,Unit3: a,G: set_a] :
( ( group_9128011940810589962sm_a_a @ Eta @ M @ Composition @ Unit @ M3 @ Composition3 @ Unit3 )
=> ( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( group_subgroup_a @ ( image_a_a @ Eta @ G ) @ M3 @ Composition3 @ Unit3 ) ) ) ).
% monoid_isomorphism.image_subgroup
thf(fact_525_SUP__identity__eq,axiom,
! [A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X3: set_a] : X3
@ A2 ) )
= ( comple2307003609928055243_set_a @ A2 ) ) ).
% SUP_identity_eq
thf(fact_526_SUP__UNION,axiom,
! [F: a > set_a,G4: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( G4 @ Y4 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_527_SUP__cong,axiom,
! [A2: set_a,B: set_a,C2: a > set_a,D2: a > set_a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_528_Sup__eqI,axiom,
! [A2: set_set_a,X: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A2 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [Y3: set_a] :
( ! [Z5: set_a] :
( ( member_set_a @ Z5 @ A2 )
=> ( ord_less_eq_set_a @ Z5 @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_529_Sup__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( ord_less_eq_set_a @ A3 @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_530_Sup__least,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_531_Sup__upper,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_532_Sup__le__iff,axiom,
! [A2: set_set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B4 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ B4 ) ) ) ) ).
% Sup_le_iff
thf(fact_533_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V2 @ U )
=> ( ord_less_eq_set_a @ V2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_534_monoid_Oinverse__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_inverse_a @ M @ Composition @ Unit )
= ( restrict_a_a
@ ^ [U2: a] :
( the_a
@ ^ [V4: a] :
( ( member_a @ V4 @ M )
& ( ( Composition @ U2 @ V4 )
= Unit )
& ( ( Composition @ V4 @ U2 )
= Unit ) ) )
@ M ) ) ) ).
% monoid.inverse_def
thf(fact_535_SUP__commute,axiom,
! [F: a > a > set_a,B: set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( F @ I ) @ B ) )
@ A2 ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_536_SUP__eq,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G4: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ ( G4 @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_537_Sup__subset__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_538_Sup__union__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_union_distrib
thf(fact_539_Fpow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_540_SUP__eqI,axiom,
! [A2: set_a,F: a > set_a,X: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_541_SUP__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G4: a > set_a] :
( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G4 @ X4 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ).
% SUP_mono
thf(fact_542_SUP__least,axiom,
! [A2: set_a,F: a > set_a,U: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_543_SUP__mono_H,axiom,
! [F: a > set_a,G4: a > set_a,A2: set_a] :
( ! [X2: a] : ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_544_SUP__upper,axiom,
! [I4: a,A2: set_a,F: a > set_a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_545_SUP__le__iff,axiom,
! [F: a > set_a,A2: set_a,U: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ U )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_546_SUP__upper2,axiom,
! [I4: a,A2: set_a,U: set_a,F: a > set_a] :
( ( member_a @ I4 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_547_SUP__absorb,axiom,
! [K: a,I5: set_a,A2: a > set_a] :
( ( member_a @ K @ I5 )
=> ( ( sup_sup_set_a @ ( A2 @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) ) ) ).
% SUP_absorb
thf(fact_548_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: a > set_a,A2: set_a,G4: a > set_a] :
( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( sup_sup_set_a @ ( F @ A4 ) @ ( G4 @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_549_Sup__inter__less__eq,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_550_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_551_SUP__union,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_552_restrict__apply,axiom,
( restrict_a_a
= ( ^ [F2: a > a,A6: set_a,X3: a] : ( if_a @ ( member_a @ X3 @ A6 ) @ ( F2 @ X3 ) @ undefined_a ) ) ) ).
% restrict_apply
thf(fact_553_FuncSet_Orestrict__restrict,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( restrict_a_a @ ( restrict_a_a @ F @ A2 ) @ B )
= ( restrict_a_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% FuncSet.restrict_restrict
thf(fact_554_image__restrict__eq,axiom,
! [F: a > set_a,A2: set_a] :
( ( image_a_set_a @ ( restrict_a_set_a @ F @ A2 ) @ A2 )
= ( image_a_set_a @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_555_image__restrict__eq,axiom,
! [F: a > a,A2: set_a] :
( ( image_a_a @ ( restrict_a_a @ F @ A2 ) @ A2 )
= ( image_a_a @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_556_the__equality,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ( the_a @ P )
= A ) ) ) ).
% the_equality
thf(fact_557_the__eq__trivial,axiom,
! [A: a] :
( ( the_a
@ ^ [X3: a] : ( X3 = A ) )
= A ) ).
% the_eq_trivial
thf(fact_558_the__sym__eq__trivial,axiom,
! [X: a] :
( ( the_a
@ ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_559_Union__Un__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Un_distrib
thf(fact_560_ball__UN,axiom,
! [B: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B @ X3 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_561_bex__UN,axiom,
! [B: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B @ X3 ) )
& ( P @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_562_UN__ball__bex__simps_I2_J,axiom,
! [B: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B @ X3 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_563_UN__ball__bex__simps_I4_J,axiom,
! [B: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B @ X3 ) )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_564_UN__I,axiom,
! [A: a,A2: set_a,B4: a,B: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ B4 @ ( B @ A ) )
=> ( member_a @ B4 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_565_UN__iff,axiom,
! [B4: a,B: a > set_a,A2: set_a] :
( ( member_a @ B4 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( member_a @ B4 @ ( B @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_566_UN__Un,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_567_SUP__Sup__eq,axiom,
! [S: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I: set_a,X3: a] : ( member_a @ X3 @ I )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_568_SUP__UN__eq,axiom,
! [R2: a > set_a,S: set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_a_a_o
@ ^ [I: a,X3: a] : ( member_a @ X3 @ ( R2 @ I ) )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_569_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_570_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S2: set_a_o,X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_571_Union__eq,axiom,
( comple7399068483239264473et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X3: nat] :
? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ A6 )
& ( member_nat @ X3 @ Y4 ) ) ) ) ) ).
% Union_eq
thf(fact_572_Union__eq,axiom,
( comple2307003609928055243_set_a
= ( ^ [A6: set_set_a] :
( collect_a
@ ^ [X3: a] :
? [Y4: set_a] :
( ( member_set_a @ Y4 @ A6 )
& ( member_a @ X3 @ Y4 ) ) ) ) ) ).
% Union_eq
thf(fact_573_Union__subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ? [Y6: set_a] :
( ( member_set_a @ Y6 @ B )
& ( ord_less_eq_set_a @ X2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_574_Union__upper,axiom,
! [B: set_a,A2: set_set_a] :
( ( member_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_575_Union__least,axiom,
! [A2: set_set_a,C2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A2 )
=> ( ord_less_eq_set_a @ X6 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_576_UN__extend__simps_I9_J,axiom,
! [C2: a > set_a,B: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( B @ X3 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_577_UN__E,axiom,
! [B4: a,B: a > set_a,A2: set_a] :
( ( member_a @ B4 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ B4 @ ( B @ X2 ) ) ) ) ).
% UN_E
thf(fact_578_UN__UN__flatten,axiom,
! [C2: a > set_a,B: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( B @ Y4 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_579_Union__Int__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_580_Union__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_581_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X3: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_582_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A6: set_set_a] :
( collect_a
@ ^ [X3: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_583_UN__extend__simps_I10_J,axiom,
! [B: set_a > set_a,F: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( B @ ( F @ A4 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_584_UN__extend__simps_I10_J,axiom,
! [B: a > set_a,F: a > a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( B @ ( F @ A4 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_585_image__UN,axiom,
! [F: a > set_a,B: a > set_a,A2: set_a] :
( ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( image_a_set_a @ F @ ( B @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_586_image__UN,axiom,
! [F: a > a,B: a > set_a,A2: set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( image_a_a @ F @ ( B @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_587_UN__subset__iff,axiom,
! [A2: a > set_a,I5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ord_less_eq_set_a @ ( A2 @ X3 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_588_UN__upper,axiom,
! [A: a,A2: set_a,B: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_589_UN__least,axiom,
! [A2: set_a,B: a > set_a,C2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_590_UN__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_591_Int__UN__distrib2,axiom,
! [A2: a > set_a,I5: set_a,B: a > set_a,J3: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ J3 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J: a] : ( inf_inf_set_a @ ( A2 @ I ) @ ( B @ J ) )
@ J3 ) )
@ I5 ) ) ) ).
% Int_UN_distrib2
thf(fact_592_Int__UN__distrib,axiom,
! [B: set_a,A2: a > set_a,I5: set_a] :
( ( inf_inf_set_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( inf_inf_set_a @ B @ ( A2 @ I ) )
@ I5 ) ) ) ).
% Int_UN_distrib
thf(fact_593_UN__extend__simps_I4_J,axiom,
! [A2: a > set_a,C2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(4)
thf(fact_594_UN__extend__simps_I5_J,axiom,
! [A2: set_a,B: a > set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ A2 @ ( B @ X3 ) )
@ C2 ) ) ) ).
% UN_extend_simps(5)
thf(fact_595_Un__Union__image,axiom,
! [A2: a > set_a,B: a > set_a,C2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ ( B @ X3 ) )
@ C2 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_596_UN__Un__distrib,axiom,
! [A2: a > set_a,B: a > set_a,I5: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( sup_sup_set_a @ ( A2 @ I ) @ ( B @ I ) )
@ I5 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ I5 ) ) ) ) ).
% UN_Un_distrib
thf(fact_597_UN__absorb,axiom,
! [K: a,I5: set_a,A2: a > set_a] :
( ( member_a @ K @ I5 )
=> ( ( sup_sup_set_a @ ( A2 @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) ) ) ).
% UN_absorb
thf(fact_598_Collect__bex__eq,axiom,
! [A2: set_a,P: a > a > $o] :
( ( collect_a
@ ^ [X3: a] :
? [Y4: a] :
( ( member_a @ Y4 @ A2 )
& ( P @ X3 @ Y4 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] :
( collect_a
@ ^ [X3: a] : ( P @ X3 @ Y4 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_599_UNION__eq,axiom,
! [B: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) )
= ( collect_a
@ ^ [Y4: a] :
? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( member_a @ Y4 @ ( B @ X3 ) ) ) ) ) ).
% UNION_eq
thf(fact_600_image__Union,axiom,
! [F: a > set_a,S: set_set_a] :
( ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple3958522678809307947_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_601_image__Union,axiom,
! [F: a > a,S: set_set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_602_Int__Union2,axiom,
! [B: set_set_a,A2: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A2 )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [C6: set_a] : ( inf_inf_set_a @ C6 @ A2 )
@ B ) ) ) ).
% Int_Union2
thf(fact_603_Int__Union,axiom,
! [A2: set_a,B: set_set_a] :
( ( inf_inf_set_a @ A2 @ ( comple2307003609928055243_set_a @ B ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A2 ) @ B ) ) ) ).
% Int_Union
thf(fact_604_UN__extend__simps_I8_J,axiom,
! [B: a > set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y4: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ Y4 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_605_restrict__apply_H,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( restrict_a_a @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_606_restrict__ext,axiom,
! [A2: set_a,F: a > a,G4: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( restrict_a_a @ F @ A2 )
= ( restrict_a_a @ G4 @ A2 ) ) ) ).
% restrict_ext
thf(fact_607_the1__equality,axiom,
! [P: a > $o,A: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A )
=> ( ( the_a @ P )
= A ) ) ) ).
% the1_equality
thf(fact_608_the1I2,axiom,
! [P: a > $o,Q: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ).
% the1I2
thf(fact_609_If__def,axiom,
( if_a
= ( ^ [P2: $o,X3: a,Y4: a] :
( the_a
@ ^ [Z6: a] :
( ( P2
=> ( Z6 = X3 ) )
& ( ~ P2
=> ( Z6 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_610_theI2,axiom,
! [P: a > $o,A: a,Q: a > $o] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ) ).
% theI2
thf(fact_611_theI_H,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( P @ ( the_a @ P ) ) ) ).
% theI'
thf(fact_612_theI,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( P @ ( the_a @ P ) ) ) ) ).
% theI
thf(fact_613_restrict__def,axiom,
( restrict_a_a
= ( ^ [F2: a > a,A6: set_a,X3: a] : ( if_a @ ( member_a @ X3 @ A6 ) @ ( F2 @ X3 ) @ undefined_a ) ) ) ).
% restrict_def
thf(fact_614_SUP__inf__distrib2,axiom,
! [F: a > set_a,A2: set_a,G4: a > set_a,B: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ ( F @ A4 ) @ ( G4 @ B3 ) )
@ B ) )
@ A2 ) ) ) ).
% SUP_inf_distrib2
thf(fact_615_inf__SUP,axiom,
! [A: set_a,F: a > set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ B ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ A @ ( F @ B3 ) )
@ B ) ) ) ).
% inf_SUP
thf(fact_616_Sup__inf,axiom,
! [B: set_set_a,A: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [B3: set_a] : ( inf_inf_set_a @ B3 @ A )
@ B ) ) ) ).
% Sup_inf
thf(fact_617_SUP__inf,axiom,
! [F: a > set_a,B: set_a,A: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ B ) ) @ A )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ ( F @ B3 ) @ A )
@ B ) ) ) ).
% SUP_inf
thf(fact_618_inf__Sup,axiom,
! [A: set_a,B: set_set_a] :
( ( inf_inf_set_a @ A @ ( comple2307003609928055243_set_a @ B ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A ) @ B ) ) ) ).
% inf_Sup
thf(fact_619_UN__constant__eq,axiom,
! [A: a,A2: set_a,F: a > set_a,C: set_a] :
( ( member_a @ A @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_620_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_621_cSup__eq__maximum,axiom,
! [Z: set_a,X5: set_set_a] :
( ( member_set_a @ Z @ X5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_622_sumset,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( insert_a @ ( addition @ A4 @ B3 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ g ) ) )
@ ( inf_inf_set_a @ A2 @ g ) ) ) ) ).
% sumset
thf(fact_623_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_624_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_625_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_626_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_627_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_628_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_629_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_630_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_631_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_632_image__is__empty,axiom,
! [F: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_633_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_634_empty__is__image,axiom,
! [F: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_635_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_636_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_637_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_638_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_639_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_640_image__insert,axiom,
! [F: a > set_a,A: a,B: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A @ B ) )
= ( insert_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_insert
thf(fact_641_image__insert,axiom,
! [F: a > a,A: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_642_insert__image,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) )
= ( image_a_set_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_643_insert__image,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_644_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_645_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_646_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_647_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_648_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_649_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_650_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_651_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_652_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_653_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_654_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_655_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_656_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_657_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_658_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_659_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_660_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_661_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_662_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_663_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_664_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_665_Un__insert__left,axiom,
! [A: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_666_bex__empty,axiom,
! [P: a > $o] :
~ ? [X4: a] :
( ( member_a @ X4 @ bot_bot_set_a )
& ( P @ X4 ) ) ).
% bex_empty
thf(fact_667_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_668_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_669_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_670_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_671_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_672_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_673_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_674_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_675_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_676_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_677_Sup__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ A2 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_insert
thf(fact_678_SUP__bot__conv_I2_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_679_SUP__bot__conv_I1_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_680_SUP__bot,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% SUP_bot
thf(fact_681_cSUP__const,axiom,
! [A2: set_a,C: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_682_SUP__const,axiom,
! [A2: set_a,F: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_683_UN__constant,axiom,
! [A2: set_a,C: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : C
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_684_UN__simps_I1_J,axiom,
! [C2: set_a,A: a,B: a > set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C2 ) )
= ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_685_UN__singleton,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ X3 @ bot_bot_set_a )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_686_UN__simps_I3_J,axiom,
! [C2: set_a,A2: set_a,B: a > set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C2 ) )
= ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_687_UN__simps_I2_J,axiom,
! [C2: set_a,A2: a > set_a,B: set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C2 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_688_UN__insert,axiom,
! [B: a > set_a,A: a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( B @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_insert
thf(fact_689_image__constant,axiom,
! [X: a,A2: set_a,C: set_a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_set_a
@ ^ [X3: a] : C
@ A2 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ).
% image_constant
thf(fact_690_image__constant,axiom,
! [X: a,A2: set_a,C: a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_a
@ ^ [X3: a] : C
@ A2 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_691_image__constant__conv,axiom,
! [A2: set_a,C: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X3: a] : C
@ A2 )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X3: a] : C
@ A2 )
= ( insert_set_a @ C @ bot_bot_set_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_692_image__constant__conv,axiom,
! [A2: set_a,C: a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X3: a] : C
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X3: a] : C
@ A2 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_693_SUP__constant,axiom,
! [A2: set_a,C: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : C
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y4: a] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_694_SUP__empty,axiom,
! [F: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% SUP_empty
thf(fact_695_UN__extend__simps_I1_J,axiom,
! [C2: set_a,A: a,B: a > set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_696_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_697_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_698_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_699_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_700_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_701_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_702_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_703_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_704_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_705_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_706_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_707_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_708_Int__insert__left,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_709_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_710_cSup__least,axiom,
! [X5: set_set_a,Z: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_711_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_712_cSup__eq__non__empty,axiom,
! [X5: set_set_a,A: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( ! [Y3: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ X5 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) )
=> ( ord_less_eq_set_a @ A @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_713_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_714_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_715_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_716_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_717_insert__subsetI,axiom,
! [X: a,A2: set_a,X5: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_718_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_719_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_720_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_721_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_722_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_723_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_724_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_725_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_726_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_727_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_728_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A4 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_729_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B6: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A4 )
| ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_730_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_731_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_732_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_733_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_734_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_735_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_736_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B8: set_a] :
( ( A2
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_737_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_738_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_739_doubleton__eq__iff,axiom,
! [A: a,B4: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_740_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_741_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_742_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C6: set_a] :
( ( A2
= ( insert_a @ B4 @ C6 ) )
& ~ ( member_a @ B4 @ C6 )
& ( B
= ( insert_a @ A @ C6 ) )
& ~ ( member_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_743_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_744_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_745_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_746_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B8: set_a] :
( ( A2
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_747_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_748_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_749_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_750_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_751_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_752_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_753_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_754_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_755_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_756_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_757_UNION__singleton__eq__range,axiom,
! [F: a > set_a,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( insert_set_a @ ( F @ X3 ) @ bot_bot_set_set_a )
@ A2 ) )
= ( image_a_set_a @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_758_UNION__singleton__eq__range,axiom,
! [F: a > a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ ( F @ X3 ) @ bot_bot_set_a )
@ A2 ) )
= ( image_a_a @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_759_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_a,A: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( inf_inf_set_a @ X3 @ A )
= bot_bot_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_760_cSUP__least,axiom,
! [A2: set_a,F: a > nat,M: nat] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_761_cSUP__least,axiom,
! [A2: set_a,F: a > set_a,M: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_762_less__eq__Sup,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V3: set_a] :
( ( member_set_a @ V3 @ A2 )
=> ( ord_less_eq_set_a @ U @ V3 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_763_SUP__eq__const,axiom,
! [I5: set_a,F: a > set_a,X: set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_764_Union__disjoint,axiom,
! [C2: set_set_a,A2: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C2 ) @ A2 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( ( inf_inf_set_a @ X3 @ A2 )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_765_UN__insert__distrib,axiom,
! [U: a,A2: set_a,A: a,B: a > set_a] :
( ( member_a @ U @ A2 )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_766_UN__empty2,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% UN_empty2
thf(fact_767_UN__empty,axiom,
! [B: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% UN_empty
thf(fact_768_UNION__empty__conv_I1_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_769_UNION__empty__conv_I2_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_770_additive__abelian__group_Osumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( insert_a @ ( Addition @ A4 @ B3 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ G ) ) )
@ ( inf_inf_set_a @ A2 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset
thf(fact_771_Union__image__insert,axiom,
! [F: a > set_a,A: a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( insert_a @ A @ B ) ) )
= ( sup_sup_set_a @ ( F @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ B ) ) ) ) ).
% Union_image_insert
thf(fact_772_SUP__eq__iff,axiom,
! [I5: set_a,C: set_a,F: a > set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ord_less_eq_set_a @ C @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_773_Union__image__empty,axiom,
! [A2: set_a,F: a > set_a] :
( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ bot_bot_set_a ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_774_SUP__insert,axiom,
! [F: a > set_a,A: a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( F @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_775_UN__extend__simps_I3_J,axiom,
! [C2: set_a,A2: set_a,B: a > set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) )
= A2 ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_776_UN__extend__simps_I2_J,axiom,
! [C2: set_a,A2: a > set_a,B: set_a] :
( ( ( C2 = bot_bot_set_a )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ B )
= B ) )
& ( ( C2 != bot_bot_set_a )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_777_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_778_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_779_the__elem__def,axiom,
( the_elem_a
= ( ^ [X7: set_a] :
( the_a
@ ^ [X3: a] :
( X7
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_780_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_781_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_782_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_783_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_784_Union__insert,axiom,
! [A: set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_insert
thf(fact_785_insert__partition,axiom,
! [X: set_a,F3: set_set_a] :
( ~ ( member_set_a @ X @ F3 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( insert_set_a @ X @ F3 ) )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( insert_set_a @ X @ F3 ) )
=> ( ( X2 != Xa )
=> ( ( inf_inf_set_a @ X2 @ Xa )
= bot_bot_set_a ) ) ) )
=> ( ( inf_inf_set_a @ X @ ( comple2307003609928055243_set_a @ F3 ) )
= bot_bot_set_a ) ) ) ).
% insert_partition
thf(fact_786_the__elem__image__unique,axiom,
! [A2: set_a,F: a > a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_787_the__elem__image__unique,axiom,
! [A2: set_a,F: a > set_a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_788_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_789_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_790_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_791_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_792_comp__fun__commute__Pow__fold,axiom,
( finite312795530508511377_set_a
@ ^ [X3: a,A6: set_set_a] : ( sup_sup_set_set_a @ A6 @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_793_comp__fun__commute__filter__fold,axiom,
! [P: a > $o] :
( finite3518785373051244337_set_a
@ ^ [X3: a,A8: set_a] : ( if_set_a @ ( P @ X3 ) @ ( insert_a @ X3 @ A8 ) @ A8 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_794_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_795_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X3: a] :
( A6
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_796_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X2: a] :
( A2
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_797_cSUP__UNION,axiom,
! [A2: set_a,B: a > set_a,F: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B @ X2 )
!= bot_bot_set_a ) )
=> ( ( condit3373647341569784514_set_a
@ ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( image_a_set_a @ F @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_798_cSUP__union,axiom,
! [A2: set_a,F: a > set_a,B: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( B != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ B ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_799_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_800_bdd__above_OI,axiom,
! [A2: set_set_a,M: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ M ) )
=> ( condit3373647341569784514_set_a @ A2 ) ) ).
% bdd_above.I
thf(fact_801_bdd__above_OI,axiom,
! [A2: set_nat,M: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_802_bdd__above__image__sup,axiom,
! [F: a > set_a,G4: a > set_a,A2: set_a] :
( ( condit3373647341569784514_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( F @ X3 ) @ ( G4 @ X3 ) )
@ A2 ) )
= ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
& ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_803_bdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A6: set_set_a] :
? [M2: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( ord_less_eq_set_a @ X3 @ M2 ) ) ) ) ).
% bdd_above.unfold
thf(fact_804_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A6: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% bdd_above.unfold
thf(fact_805_bdd__above_OE,axiom,
! [A2: set_set_a] :
( ( condit3373647341569784514_set_a @ A2 )
=> ~ ! [M4: set_a] :
~ ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ X4 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_806_bdd__above_OE,axiom,
! [A2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ~ ! [M4: nat] :
~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_807_bdd__above_OI2,axiom,
! [A2: set_a,F: a > set_a,M: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_808_bdd__above_OI2,axiom,
! [A2: set_a,F: a > nat,M: nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_809_cSup__upper,axiom,
! [X: nat,X5: set_nat] :
( ( member_nat @ X @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_810_cSup__upper,axiom,
! [X: set_a,X5: set_set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( condit3373647341569784514_set_a @ X5 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_811_cSup__upper2,axiom,
! [X: nat,X5: set_nat,Y: nat] :
( ( member_nat @ X @ X5 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_812_cSup__upper2,axiom,
! [X: set_a,X5: set_set_a,Y: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( ( condit3373647341569784514_set_a @ X5 )
=> ( ord_less_eq_set_a @ Y @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_813_cSUP__upper2,axiom,
! [F: a > nat,A2: set_a,X: a,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) )
=> ( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_814_cSUP__upper2,axiom,
! [F: a > set_a,A2: set_a,X: a,U: set_a] :
( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ X ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_815_cSUP__upper,axiom,
! [X: a,A2: set_a,F: a > nat] :
( ( member_a @ X @ A2 )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_816_cSUP__upper,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_817_cSup__le__iff,axiom,
! [S: set_nat,A: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( ord_less_eq_nat @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_818_cSup__le__iff,axiom,
! [S: set_set_a,A: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S ) @ A )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_819_cSup__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( B != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ B2 @ X4 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B ) @ ( complete_Sup_Sup_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_820_cSup__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A2 )
=> ( ! [B2: set_a] :
( ( member_set_a @ B2 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ B2 @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B ) @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_821_cSUP__mono,axiom,
! [A2: set_a,G4: a > set_a,B: set_a,F: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ B ) )
=> ( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G4 @ X4 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_822_cSUP__le__iff,axiom,
! [A2: set_a,F: a > nat,U: nat] :
( ( A2 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) @ U )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_823_cSUP__le__iff,axiom,
! [A2: set_a,F: a > set_a,U: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ U )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_824_cSup__subset__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_825_cSup__subset__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_826_cSup__insert__If,axiom,
! [X5: set_set_a,A: set_a] :
( ( condit3373647341569784514_set_a @ X5 )
=> ( ( ( X5 = bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X5 ) )
= A ) )
& ( ( X5 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X5 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_827_cSup__insert,axiom,
! [X5: set_set_a,A: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ X5 )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X5 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ) ).
% cSup_insert
thf(fact_828_cSup__union__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_829_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_a,F: a > set_a,G4: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ A2 ) )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( sup_sup_set_a @ ( F @ A4 ) @ ( G4 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_830_cSUP__subset__mono,axiom,
! [A2: set_a,G4: a > nat,B: set_a,F: a > nat] :
( ( A2 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ G4 @ B ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ G4 @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_831_cSUP__subset__mono,axiom,
! [A2: set_a,G4: a > set_a,B: set_a,F: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ B ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_832_cSup__inter__less__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( condit2214826472909112428ve_nat @ B )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_833_cSup__inter__less__eq,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( condit3373647341569784514_set_a @ A2 )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A2 @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_834_cSUP__insert,axiom,
! [A2: set_a,F: a > set_a,A: a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( F @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_835_subset__singleton__iff__Uniq,axiom,
! [A2: set_a] :
( ( ? [A4: a] : ( ord_less_eq_set_a @ A2 @ ( insert_a @ A4 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_836_bind__singleton__conv__image,axiom,
! [A2: set_a,F: a > set_a] :
( ( bind_a_set_a @ A2
@ ^ [X3: a] : ( insert_set_a @ ( F @ X3 ) @ bot_bot_set_set_a ) )
= ( image_a_set_a @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_837_bind__singleton__conv__image,axiom,
! [A2: set_a,F: a > a] :
( ( bind_a_a @ A2
@ ^ [X3: a] : ( insert_a @ ( F @ X3 ) @ bot_bot_set_a ) )
= ( image_a_a @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_838_less__cSUP__iff,axiom,
! [A2: set_a,F: a > nat,A: nat] :
( ( A2 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) )
=> ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( ord_less_nat @ A @ ( F @ X3 ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_839_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_840_empty__bind,axiom,
! [F: a > set_a] :
( ( bind_a_a @ bot_bot_set_a @ F )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_841_member__bind,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ ( bind_a_a @ A2 @ F ) )
= ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_842_inf_Ostrict__coboundedI2,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_843_inf_Ostrict__coboundedI2,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_844_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_845_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B4: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_846_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_847_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_848_inf_Ostrict__boundedE,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B4 @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B4 )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_849_inf_Ostrict__boundedE,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B4 @ C ) )
=> ~ ( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_850_inf_Oabsorb4,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_851_inf_Oabsorb4,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_852_inf_Oabsorb3,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_853_inf_Oabsorb3,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_854_less__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_855_less__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_nat @ B4 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_856_less__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_857_less__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_858_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B4: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B4 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_859_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B4: nat,A: nat] :
( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_860_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B4: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_861_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B4: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_862_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_863_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_864_sup_Ostrict__boundedE,axiom,
! [B4: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B4 @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B4 @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_865_sup_Ostrict__boundedE,axiom,
! [B4: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B4 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_866_sup_Oabsorb4,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb4
thf(fact_867_sup_Oabsorb4,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb4
thf(fact_868_sup_Oabsorb3,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb3
thf(fact_869_sup_Oabsorb3,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb3
thf(fact_870_less__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B4 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% less_supI2
thf(fact_871_less__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_nat @ X @ B4 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% less_supI2
thf(fact_872_less__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% less_supI1
thf(fact_873_less__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% less_supI1
thf(fact_874_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A9: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A9 ) )
= ( ord_less_nat @ A9 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_875_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_876_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_877_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_878_nless__le,axiom,
! [A: set_a,B4: set_a] :
( ( ~ ( ord_less_set_a @ A @ B4 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_879_nless__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_880_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_881_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_882_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_883_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_884_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_885_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_886_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_887_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_888_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_889_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_890_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_891_order_Ostrict__trans1,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ B4 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_892_order_Ostrict__trans1,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_893_order_Ostrict__trans2,axiom,
! [A: set_a,B4: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_894_order_Ostrict__trans2,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_895_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_896_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_897_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_898_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_899_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_900_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_901_dual__order_Ostrict__trans1,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_set_a @ C @ B4 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_902_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_903_dual__order_Ostrict__trans2,axiom,
! [B4: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C @ B4 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_904_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A: nat,C: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_905_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_906_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_907_order_Ostrict__implies__order,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_908_order_Ostrict__implies__order,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_909_dual__order_Ostrict__implies__order,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_910_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_911_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_912_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_913_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_914_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_915_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_916_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_917_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_918_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_919_order__le__neq__trans,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_920_order__le__neq__trans,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_921_order__neq__le__trans,axiom,
! [A: set_a,B4: set_a] :
( ( A != B4 )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_922_order__neq__le__trans,axiom,
! [A: nat,B4: nat] :
( ( A != B4 )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_923_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_924_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_925_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_926_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_927_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_928_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_929_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ ( F @ B4 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_930_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_931_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_set_a @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_932_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_933_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_934_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_935_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_936_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_937_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_938_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_939_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_940_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_941_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_942_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_943_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_944_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_945_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_946_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_947_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_948_order__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_949_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_950_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_951_ord__less__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_952_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_953_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_954_order__less__asym_H,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order_less_asym'
thf(fact_955_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_956_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_957_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_958_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( A != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_959_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( A != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_960_dual__order_Ostrict__trans,axiom,
! [B4: nat,A: nat,C: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_961_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_962_order_Ostrict__trans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_963_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_964_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P2: nat > $o] :
? [N3: nat] :
( ( P2 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_965_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_966_dual__order_Oasym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ A @ B4 ) ) ).
% dual_order.asym
thf(fact_967_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_968_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_969_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_970_ord__less__eq__trans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( B4 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_971_ord__eq__less__trans,axiom,
! [A: nat,B4: nat,C: nat] :
( ( A = B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_972_order_Oasym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order.asym
thf(fact_973_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_974_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_975_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_976_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_977_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_978_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_979_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_980_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_981_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_982_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_983_complete__interval,axiom,
! [A: nat,B4: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B4 )
=> ( ( P @ A )
=> ( ~ ( P @ B4 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B4 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_984_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_985_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_986_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_987_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_988_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_989_bind__const,axiom,
! [A2: set_a,B: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( bind_a_a @ A2
@ ^ [Uu: a] : B )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( bind_a_a @ A2
@ ^ [Uu: a] : B )
= B ) ) ) ).
% bind_const
thf(fact_990_the1__equality_H,axiom,
! [P: a > $o,A: a] :
( ( uniq_a @ P )
=> ( ( P @ A )
=> ( ( the_a @ P )
= A ) ) ) ).
% the1_equality'
thf(fact_991_SUP__lessD,axiom,
! [F: a > set_a,A2: set_a,Y: set_a,I4: a] :
( ( ord_less_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ Y )
=> ( ( member_a @ I4 @ A2 )
=> ( ord_less_set_a @ ( F @ I4 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_992_bind__UNION,axiom,
( bind_a_a
= ( ^ [A6: set_a,F2: a > set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F2 @ A6 ) ) ) ) ).
% bind_UNION
thf(fact_993_cSUP__lessD,axiom,
! [F: a > nat,A2: set_a,Y: nat,I4: a] :
( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A2 ) )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) @ Y )
=> ( ( member_a @ I4 @ A2 )
=> ( ord_less_nat @ ( F @ I4 ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_994_cSUP__lessD,axiom,
! [F: a > set_a,A2: set_a,Y: set_a,I4: a] :
( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ( ( ord_less_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ Y )
=> ( ( member_a @ I4 @ A2 )
=> ( ord_less_set_a @ ( F @ I4 ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_995_Set_Obind__def,axiom,
( bind_a_a
= ( ^ [A6: set_a,F2: a > set_a] :
( collect_a
@ ^ [X3: a] :
? [Y4: set_a] :
( ( member_set_a @ Y4 @ ( image_a_set_a @ F2 @ A6 ) )
& ( member_a @ X3 @ Y4 ) ) ) ) ) ).
% Set.bind_def
thf(fact_996_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2496817875450240012_set_a @ sup_sup_set_a @ bot_bot_set_a
@ ^ [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X3 )
@ ^ [X3: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_997_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_998_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_999_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1000_psubsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1001_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_1002_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_1003_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X3 )
@ ^ [X3: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X3 )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_1004_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X3 )
@ ^ [X3: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X3 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_1005_image__Pow__mono,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A2 ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_1006_image__Pow__mono,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A2 ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_1007_Pow__iff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Pow_iff
thf(fact_1008_PowI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_1009_Pow__singleton__iff,axiom,
! [X5: set_a,Y5: set_a] :
( ( ( pow_a @ X5 )
= ( insert_set_a @ Y5 @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_a )
& ( Y5 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1010_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_1011_Pow__Int__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pow_a @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_1012_Pow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_1013_image__Pow__surj,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( ( image_a_a @ F @ A2 )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A2 ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_1014_image__Pow__surj,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ( image_a_set_a @ F @ A2 )
= B )
=> ( ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A2 ) )
= ( pow_set_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_1015_Pow__def,axiom,
( pow_a
= ( ^ [A6: set_a] :
( collect_set_a
@ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_1016_PowD,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% PowD
thf(fact_1017_Cantors__paradox,axiom,
! [A2: set_a] :
~ ? [F4: a > set_a] :
( ( image_a_set_a @ F4 @ A2 )
= ( pow_a @ A2 ) ) ).
% Cantors_paradox
thf(fact_1018_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_1019_Un__Pow__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) @ ( pow_a @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_Pow_subset
thf(fact_1020_Pow__insert,axiom,
! [A: a,A2: set_a] :
( ( pow_a @ ( insert_a @ A @ A2 ) )
= ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( image_set_a_set_a @ ( insert_a @ A ) @ ( pow_a @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_1021_UN__Pow__subset,axiom,
! [B: a > set_a,A2: set_a] :
( ord_le3724670747650509150_set_a
@ ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( pow_a @ ( B @ X3 ) )
@ A2 ) )
@ ( pow_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_Pow_subset
thf(fact_1022_partition_Ointro,axiom,
! [P: set_set_a,S: set_a] :
( ( ord_le3724670747650509150_set_a @ P @ ( pow_a @ S ) )
=> ( ~ ( member_set_a @ bot_bot_set_a @ P )
=> ( ( ( comple2307003609928055243_set_a @ P )
= S )
=> ( ! [A10: set_a,B8: set_a] :
( ( member_set_a @ A10 @ P )
=> ( ( member_set_a @ B8 @ P )
=> ( ( A10 != B8 )
=> ( ( inf_inf_set_a @ A10 @ B8 )
= bot_bot_set_a ) ) ) )
=> ( set_partition_a @ S @ P ) ) ) ) ) ).
% partition.intro
thf(fact_1023_Set__Theory_Opartition__def,axiom,
( set_partition_a
= ( ^ [S2: set_a,P2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ P2 @ ( pow_a @ S2 ) )
& ~ ( member_set_a @ bot_bot_set_a @ P2 )
& ( ( comple2307003609928055243_set_a @ P2 )
= S2 )
& ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ P2 )
=> ( ( member_set_a @ B6 @ P2 )
=> ( ( A6 != B6 )
=> ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% Set_Theory.partition_def
thf(fact_1024_partition_Odisjoint,axiom,
! [S: set_a,P: set_set_a,A2: set_a,B: set_a] :
( ( set_partition_a @ S @ P )
=> ( ( member_set_a @ A2 @ P )
=> ( ( member_set_a @ B @ P )
=> ( ( A2 != B )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ) ) ).
% partition.disjoint
thf(fact_1025_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1026_Fpow__Pow__finite,axiom,
( finite_Fpow_nat
= ( ^ [A6: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A6 ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1027_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1028_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1029_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
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1030_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
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1031_finite__imageI,axiom,
! [F3: set_a,H3: a > a] :
( ( finite_finite_a @ F3 )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F3 ) ) ) ).
% finite_imageI
thf(fact_1032_finite__imageI,axiom,
! [F3: set_a,H3: a > set_a] :
( ( finite_finite_a @ F3 )
=> ( finite_finite_set_a @ ( image_a_set_a @ H3 @ F3 ) ) ) ).
% finite_imageI
thf(fact_1033_finite__imageI,axiom,
! [F3: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F3 ) ) ) ).
% finite_imageI
thf(fact_1034_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_1035_finite__Int,axiom,
! [F3: set_a,G: set_a] :
( ( ( finite_finite_a @ F3 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_1036_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_1037_finite__Un,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) )
= ( ( finite_finite_a @ F3 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_1038_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1039_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1040_finite__Collect__bex,axiom,
! [A2: set_nat,Q: a > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
? [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
& ( Q @ X3 @ Y4 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y4: a] : ( Q @ Y4 @ X3 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1041_finite__Collect__bex,axiom,
! [A2: set_nat,Q: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
? [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
& ( Q @ X3 @ Y4 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y4: nat] : ( Q @ Y4 @ X3 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1042_finite__UN,axiom,
! [A2: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_nat @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1043_finite__UN,axiom,
! [A2: set_a,B: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1044_finite__UN,axiom,
! [A2: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1045_finite__UN__I,axiom,
! [A2: set_a,B: a > set_nat] :
( ( finite_finite_a @ A2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( finite_finite_nat @ ( B @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_1046_finite__UN__I,axiom,
! [A2: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( finite_finite_nat @ ( B @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_1047_finite__UN__I,axiom,
! [A2: set_a,B: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_1048_finite__UN__I,axiom,
! [A2: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_1049_nat__seg__image__imp__finite,axiom,
! [A2: set_nat,F: nat > nat,N4: nat] :
( ( A2
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) )
=> ( finite_finite_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_1050_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
? [N3: nat,F2: nat > nat] :
( A6
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1051_finite__subset__Union,axiom,
! [A2: set_nat,B10: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B10 ) )
=> ~ ! [F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ B10 )
=> ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1052_finite__subset__Union,axiom,
! [A2: set_a,B10: set_set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ B10 ) )
=> ~ ! [F5: set_set_a] :
( ( finite_finite_set_a @ F5 )
=> ( ( ord_le3724670747650509150_set_a @ F5 @ B10 )
=> ~ ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1053_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1054_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1055_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1056_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1057_Zorn__Lemma,axiom,
! [A2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A2 ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X2 ) @ A2 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1058_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1059_all__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1060_all__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1061_all__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1062_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1063_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1064_ex__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1065_ex__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1066_ex__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1067_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1068_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1069_finite__subset__image,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_set_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1070_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A2: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1071_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A2: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1072_finite__subset__image,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1073_finite__surj,axiom,
! [A2: set_a,B: set_set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B ) ) ) ).
% finite_surj
thf(fact_1074_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1075_finite__surj,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1076_finite__surj,axiom,
! [A2: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1077_le__cSup__finite,axiom,
! [X5: set_nat,X: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1078_le__cSup__finite,axiom,
! [X5: set_set_a,X: set_a] :
( ( finite_finite_set_a @ X5 )
=> ( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1079_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1080_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1081_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1082_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1083_finite__UnI,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1084_finite__UnI,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1085_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1086_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1087_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1088_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_1089_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1090_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1091_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1092_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1093_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X: set_a,Y: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X @ C )
=> ( ( member_set_a @ Y @ C )
=> ( ( ord_less_eq_set_a @ X @ Y )
| ( ord_less_eq_set_a @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_1094_Zorn__Lemma2,axiom,
! [A2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A2 ) )
=> ? [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa2 ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1095_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1096_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1097_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1098_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1099_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1100_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1101_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_1102_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_1103_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1104_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1105_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A6: set_nat] :
( collect_set_nat
@ ^ [X7: set_nat] :
( ( ord_less_eq_set_nat @ X7 @ A6 )
& ( finite_finite_nat @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_1106_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A6: set_a] :
( collect_set_a
@ ^ [X7: set_a] :
( ( ord_less_eq_set_a @ X7 @ A6 )
& ( finite_finite_a @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_1107_finite__subset__induct_H,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F6 @ A2 )
=> ( ~ ( member_nat @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1108_finite__subset__induct_H,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F6 @ A2 )
=> ( ~ ( member_a @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ A3 @ F6 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1109_finite__subset__induct,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1110_finite__subset__induct,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ A3 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1111_finite__Sup__in,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( member_set_a @ Y3 @ A2 )
=> ( member_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ A2 ) ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_1112_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,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1113_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,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1114_Pow__fold,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( pow_nat @ A2 )
= ( finite4178521680790401110et_nat
@ ^ [X3: nat,A6: set_set_nat] : ( sup_sup_set_set_nat @ A6 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ A6 ) )
@ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
@ A2 ) ) ) ).
% Pow_fold
thf(fact_1115_Pow__fold,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( pow_a @ A2 )
= ( finite9006272623207878408_set_a
@ ^ [X3: a,A6: set_set_a] : ( sup_sup_set_set_a @ A6 @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ A6 ) )
@ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a )
@ A2 ) ) ) ).
% Pow_fold
thf(fact_1116_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1117_union__fold__insert,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( finite5529483035118572448et_nat @ insert_nat @ B @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1118_union__fold__insert,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( finite_fold_a_set_a @ insert_a @ B @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1119_sup__Sup__fold__sup,axiom,
! [A2: set_set_a,B: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ B @ A2 ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1120_Sup__fold__sup,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( comple2307003609928055243_set_a @ A2 )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ bot_bot_set_a @ A2 ) ) ) ).
% Sup_fold_sup
thf(fact_1121_image__fold__insert,axiom,
! [A2: set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( image_a_set_a @ F @ A2 )
= ( finite9006272623207878408_set_a
@ ^ [K3: a] : ( insert_set_a @ ( F @ K3 ) )
@ bot_bot_set_set_a
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1122_image__fold__insert,axiom,
! [A2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( image_a_a @ F @ A2 )
= ( finite_fold_a_set_a
@ ^ [K3: a] : ( insert_a @ ( F @ K3 ) )
@ bot_bot_set_a
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1123_image__fold__insert,axiom,
! [A2: set_nat,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( image_nat_a @ F @ A2 )
= ( finite4864421574810880708_set_a
@ ^ [K3: nat] : ( insert_a @ ( F @ K3 ) )
@ bot_bot_set_a
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1124_Set__filter__fold,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( filter_nat @ P @ A2 )
= ( finite5529483035118572448et_nat
@ ^ [X3: nat,A8: set_nat] : ( if_set_nat @ ( P @ X3 ) @ ( insert_nat @ X3 @ A8 ) @ A8 )
@ bot_bot_set_nat
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_1125_Set__filter__fold,axiom,
! [A2: set_a,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( filter_a @ P @ A2 )
= ( finite_fold_a_set_a
@ ^ [X3: a,A8: set_a] : ( if_set_a @ ( P @ X3 ) @ ( insert_a @ X3 @ A8 ) @ A8 )
@ bot_bot_set_a
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_1126_member__filter,axiom,
! [X: a,P: a > $o,A2: set_a] :
( ( member_a @ X @ ( filter_a @ P @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1127_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_1128_Bex__fold,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
= ( finite_fold_nat_o
@ ^ [K3: nat,S4: $o] :
( S4
| ( P @ K3 ) )
@ $false
@ A2 ) ) ) ).
% Bex_fold
thf(fact_1129_Set_Ofilter__def,axiom,
( filter_a
= ( ^ [P2: a > $o,A6: set_a] :
( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A6 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1130_Set_Ofilter__def,axiom,
( filter_nat
= ( ^ [P2: nat > $o,A6: set_nat] :
( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A6 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1131_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C6 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ C6 )
=> ( ( ord_less_eq_set_a @ X3 @ Y4 )
| ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1132_inter__Set__filter,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( filter_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_1133_inter__Set__filter,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( filter_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_1134_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I4 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1135_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1136_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1137_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1138_DiffI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_1139_Diff__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1140_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_1141_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1142_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1143_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_1144_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_1145_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1146_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_1147_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1148_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1149_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1150_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1151_DiffE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1152_DiffD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1153_DiffD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_1154_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ~ ( member_a @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1155_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ~ ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1156_Un__Diff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C2 ) @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1157_Int__Diff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1158_Diff__Int2,axiom,
! [A2: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1159_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_1160_Diff__Int__distrib,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A2 ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1161_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1162_Diff__Un,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1163_Diff__Int,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1164_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1165_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1166_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1167_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1168_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1169_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1170_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_1171_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1172_UN__extend__simps_I6_J,axiom,
! [A2: a > set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C2 ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ ( A2 @ X3 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_1173_psubset__imp__ex__mem,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1174_image__diff__subset,axiom,
! [F: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1175_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1176_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1177_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1178_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C2 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1179_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1180_double__diff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1181_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1182_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1183_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1184_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1185_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A2: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1186_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A10: set_nat] :
( ( finite_finite_nat @ A10 )
=> ( ( A10 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A10 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A10 )
=> ( P @ ( minus_minus_set_nat @ A10 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1187_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A10: set_a] :
( ( finite_finite_a @ A10 )
=> ( ( A10 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A10 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A10 )
=> ( P @ ( minus_minus_set_a @ A10 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1188_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A10: set_nat] :
( ( finite_finite_nat @ A10 )
=> ( ( A10 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A10 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A10 )
=> ( P @ ( minus_minus_set_nat @ A10 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1189_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A10: set_a] :
( ( finite_finite_a @ A10 )
=> ( ( A10 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A10 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A10 )
=> ( P @ ( minus_minus_set_a @ A10 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1190_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A2 @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1191_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1192_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1193_UNION__fun__upd,axiom,
! [A2: a > set_a,I4: a,B: set_a,J3: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( fun_upd_a_set_a @ A2 @ I4 @ B ) @ J3 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ ( minus_minus_set_a @ J3 @ ( insert_a @ I4 @ bot_bot_set_a ) ) ) ) @ ( if_set_a @ ( member_a @ I4 @ J3 ) @ B @ bot_bot_set_a ) ) ) ).
% UNION_fun_upd
thf(fact_1194_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > set_a,Y: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_set_a @ ( fun_upd_a_set_a @ F @ X @ Y ) @ A2 )
= ( insert_set_a @ Y @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_set_a @ ( fun_upd_a_set_a @ F @ X @ Y ) @ A2 )
= ( image_a_set_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1195_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > a,Y: a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X @ Y ) @ A2 )
= ( image_a_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1196_Finite__Set_Ofold__def,axiom,
( finite_fold_nat_a
= ( ^ [F2: nat > a > a,Z6: a,A6: set_nat] : ( if_a @ ( finite_finite_nat @ A6 ) @ ( the_a @ ( finite9142365241556460134_nat_a @ F2 @ Z6 @ A6 ) ) @ Z6 ) ) ) ).
% Finite_Set.fold_def
thf(fact_1197_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A6: set_a] : ( minus_minus_set_a @ A6 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1198_flat__lub__def,axiom,
( partial_flat_lub_a
= ( ^ [B3: a,A6: set_a] :
( if_a @ ( ord_less_eq_set_a @ A6 @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ B3
@ ( the_a
@ ^ [X3: a] : ( member_a @ X3 @ ( minus_minus_set_a @ A6 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1199_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1200_Sup__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1201_Sup__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1202_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1203_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1204_inf__Sup__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1205_inf__Sup__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1206_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1207_Sup__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1208_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1209_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1210_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1211_Sup__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1212_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1213_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1214_Sup__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1215_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_nat @ A3 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1216_Sup__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
=> ! [A11: set_a] :
( ( member_set_a @ A11 @ A2 )
=> ( ord_less_eq_set_a @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1217_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A11: nat] :
( ( member_nat @ A11 @ A2 )
=> ( ord_less_eq_nat @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1218_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1219_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1220_Sup__fin_Ohom__commute,axiom,
! [H3: nat > nat,N: set_nat] :
( ! [X2: nat,Y3: nat] :
( ( H3 @ ( sup_sup_nat @ X2 @ Y3 ) )
= ( sup_sup_nat @ ( H3 @ X2 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H3 @ ( lattic1093996805478795353in_nat @ N ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H3 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1221_Sup__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N: set_set_a] :
( ! [X2: set_a,Y3: set_a] :
( ( H3 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_a @ ( H3 @ X2 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N )
=> ( ( N != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic2918178356826803221_set_a @ N ) )
= ( lattic2918178356826803221_set_a @ ( image_set_a_set_a @ H3 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1222_Sup__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1223_Sup__fin_Osubset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1224_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1225_Sup__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y3: set_a] : ( member_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( insert_set_a @ X2 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1226_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1227_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1228_Sup__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1229_Sup__fin_Ounion,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1230_Sup__fin_Oeq__fold,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X @ A2 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1231_Sup__fin_Oeq__fold,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ X @ A2 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1232_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1233_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1234_Inf__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1235_Inf__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1236_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1237_sup__Inf__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1238_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1239_Inf__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1240_Inf__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1241_Inf__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1242_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1243_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1244_Inf__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
=> ! [A11: set_a] :
( ( member_set_a @ A11 @ A2 )
=> ( ord_less_eq_set_a @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1245_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A11: nat] :
( ( member_nat @ A11 @ A2 )
=> ( ord_less_eq_nat @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1246_Inf__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ X @ A3 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1247_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_nat @ X @ A3 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1248_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1249_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1250_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1251_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1252_Inf__fin_Ohom__commute,axiom,
! [H3: nat > nat,N: set_nat] :
( ! [X2: nat,Y3: nat] :
( ( H3 @ ( inf_inf_nat @ X2 @ Y3 ) )
= ( inf_inf_nat @ ( H3 @ X2 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H3 @ ( lattic5238388535129920115in_nat @ N ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H3 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1253_Inf__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N: set_set_a] :
( ! [X2: set_a,Y3: set_a] :
( ( H3 @ ( inf_inf_set_a @ X2 @ Y3 ) )
= ( inf_inf_set_a @ ( H3 @ X2 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N )
=> ( ( N != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic8209813465164889211_set_a @ N ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H3 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1254_Inf__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1255_Inf__fin_Osubset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1256_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1257_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1258_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y3: nat] : ( member_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1259_Inf__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y3: set_a] : ( member_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( insert_set_a @ X2 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1260_Inf__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1261_Inf__fin_Ounion,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic8209813465164889211_set_a @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1262_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1263_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1264_Inf__fin_Oeq__fold,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X @ A2 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1265_Inf__fin_Oeq__fold,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite5985231929012247624_set_a @ inf_inf_set_a @ X @ A2 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1266_SUP__fold__sup,axiom,
! [A2: set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( finite_fold_a_set_a @ ( comp_s1103547301056249180et_a_a @ sup_sup_set_a @ F ) @ bot_bot_set_a @ A2 ) ) ) ).
% SUP_fold_sup
thf(fact_1267_SUP__fold__sup,axiom,
! [A2: set_nat,F: nat > set_a] :
( ( finite_finite_nat @ A2 )
=> ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A2 ) )
= ( finite4864421574810880708_set_a @ ( comp_s4757170965337636338_a_nat @ sup_sup_set_a @ F ) @ bot_bot_set_a @ A2 ) ) ) ).
% SUP_fold_sup
thf(fact_1268_extensional__insert__undefined,axiom,
! [A: a > a,I4: a,I5: set_a] :
( ( member_a_a @ A @ ( extensional_a_a @ ( insert_a @ I4 @ I5 ) ) )
=> ( member_a_a @ ( fun_upd_a_a @ A @ I4 @ undefined_a ) @ ( extensional_a_a @ I5 ) ) ) ).
% extensional_insert_undefined
thf(fact_1269_restrict__extensional__sub,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( member_a_a @ ( restrict_a_a @ F @ A2 ) @ ( extensional_a_a @ B ) ) ) ).
% restrict_extensional_sub
thf(fact_1270_extensional__empty,axiom,
( ( extensional_a_a @ bot_bot_set_a )
= ( insert_a_a
@ ^ [X3: a] : undefined_a
@ bot_bot_set_a_a ) ) ).
% extensional_empty
thf(fact_1271_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G4: set_a > a,F: a > set_a,A2: set_a] :
( ( Sup @ ( image_set_a_a @ G4 @ ( image_a_set_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_a @ ( comp_set_a_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_1272_Sup_OSUP__image,axiom,
! [Sup: set_set_a > set_a,G4: set_a > set_a,F: a > set_a,A2: set_a] :
( ( Sup @ ( image_set_a_set_a @ G4 @ ( image_a_set_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_set_a @ ( comp_set_a_set_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_1273_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G4: a > a,F: a > a,A2: set_a] :
( ( Sup @ ( image_a_a @ G4 @ ( image_a_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_a @ ( comp_a_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_1274_Sup_OSUP__image,axiom,
! [Sup: set_set_a > set_a,G4: a > set_a,F: a > a,A2: set_a] :
( ( Sup @ ( image_a_set_a @ G4 @ ( image_a_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_set_a @ ( comp_a_set_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_1275_Inf_OINF__image,axiom,
! [Inf: set_set_a > set_a,G4: set_a > set_a,F: a > set_a,A2: set_a] :
( ( Inf @ ( image_set_a_set_a @ G4 @ ( image_a_set_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_set_a @ ( comp_set_a_set_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_1276_Inf_OINF__image,axiom,
! [Inf: set_a > a,G4: a > a,F: a > a,A2: set_a] :
( ( Inf @ ( image_a_a @ G4 @ ( image_a_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_a @ ( comp_a_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_1277_Inf_OINF__image,axiom,
! [Inf: set_set_a > set_a,G4: a > set_a,F: a > a,A2: set_a] :
( ( Inf @ ( image_a_set_a @ G4 @ ( image_a_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_set_a @ ( comp_a_set_a_a @ G4 @ F ) @ A2 ) ) ) ).
% Inf.INF_image
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( sup_sup_set_a @ b @ b2 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b2 ) ) ) ).
%------------------------------------------------------------------------------