TPTP Problem File: SLH0813^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00250_008618__11856070_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1469 ( 359 unt; 198 typ; 0 def)
% Number of atoms : 4513 ( 979 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11944 ( 340 ~; 48 |; 236 &;9135 @)
% ( 0 <=>;2185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 1343 (1343 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 14 con; 0-4 aty)
% Number of variables : 3606 ( 153 ^;3305 !; 148 ?;3606 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:57:37.984
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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% Explicit typings (172)
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
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_Map_Odom_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
dom_nat_list_o: ( nat > option_list_o ) > set_nat ).
thf(sy_c_Map_Odom_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__List__Olist_I_Eo_J,type,
dom_op3781509917836105484list_o: ( option_list_o > option_list_o ) > set_option_list_o ).
thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_I_Eo_J,type,
dom_Pr5202128127435093295list_o: ( product_prod_nat_nat > option_list_o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Map_Odom_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_I_Eo_J,type,
dom_set_nat_list_o: ( set_nat > option_list_o ) > set_set_nat ).
thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
graph_nat_list_o: ( nat > option_list_o ) > set_Pr6702978257058677881list_o ).
thf(sy_c_Map_Omap__add_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
map_add_nat_list_o: ( nat > option_list_o ) > ( nat > option_list_o ) > nat > option_list_o ).
thf(sy_c_Map_Omap__le_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
map_le_nat_list_o: ( nat > option_list_o ) > ( nat > option_list_o ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
compow8708494347934031032et_nat: nat > ( set_nat > set_nat ) > set_nat > set_nat ).
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_001t__Nat__Onat,type,
bot_bot_nat: nat ).
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_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le1188267648640031866_ereal: extended_ereal > extended_ereal > $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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le5321083090456148570_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
ord_le4476516537835661936list_o: set_option_list_o > set_option_list_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
ord_le1792839605950587050on_nat: set_option_nat > set_option_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le2904074325318523657at_nat: set_Sum_sum_nat_nat > set_Sum_sum_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le1644982726543182158_ereal: set_Extended_ereal > set_Extended_ereal > $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__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
ord_le1162937763994921316list_o: set_option_list_o > set_option_list_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
ord_le6937355464348597430on_nat: set_option_nat > set_option_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
ord_le5287700718633833262_ereal: set_se6634062954251873166_ereal > set_se6634062954251873166_ereal > $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 ).
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ord_le5967974642961909525at_nat: set_Sum_sum_nat_nat > set_Sum_sum_nat_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
top_to633166595683317524list_o: set_option_list_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
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top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Prefix__Free__Code__Combinators_ONb_092_060_094sub_062e,type,
prefix6319276831915272717e_Nb_e: nat > nat > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oencode__dependent__prod_001t__Nat__Onat_001t__Nat__Onat,type,
prefix1356340002065705634at_nat: ( nat > option_list_o ) > ( nat > nat > option_list_o ) > product_prod_nat_nat > option_list_o ).
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prefix5760520046686399407at_nat: ( nat > option_list_o ) > ( nat > product_prod_nat_nat > option_list_o ) > produc7248412053542808358at_nat > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oencode__dependent__prod_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
prefix5571864788691152052_o_nat: ( option_list_o > option_list_o ) > ( option_list_o > nat > option_list_o ) > produc962871650093274083_o_nat > option_list_o ).
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prefix5045777197273762589at_nat: ( option_list_o > option_list_o ) > ( option_list_o > product_prod_nat_nat > option_list_o ) > produc3925082498052752836at_nat > option_list_o ).
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prefix2400472653643684305at_nat: ( product_prod_nat_nat > option_list_o ) > ( product_prod_nat_nat > nat > option_list_o ) > produc8373899037510109440at_nat > option_list_o ).
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prefix8375143551930000832at_nat: ( product_prod_nat_nat > option_list_o ) > ( product_prod_nat_nat > product_prod_nat_nat > option_list_o ) > produc859450856879609959at_nat > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oencode__dependent__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
prefix3505681232979180120at_nat: ( set_nat > option_list_o ) > ( set_nat > nat > option_list_o ) > produc7491599851749785783at_nat > option_list_o ).
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prefix1222217955897696377at_nat: ( set_nat > option_list_o ) > ( set_nat > product_prod_nat_nat > option_list_o ) > produc7312547201344536560at_nat > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Nat__Onat,type,
prefix3558185134189398382ng_nat: ( nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
prefix6898455076116784582list_o: ( option_list_o > option_list_o ) > $o ).
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prefix5049516368958873059at_nat: ( product_prod_nat_nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
prefix726093921677949892at_nat: ( produc7248412053542808358at_nat > option_list_o ) > $o ).
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prefix6912799275947769473_o_nat: ( produc962871650093274083_o_nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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prefix1851580905645250974at_nat: ( produc8373899037510109440at_nat > option_list_o ) > $o ).
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prefix8803165839153646153at_nat: ( produc859450856879609959at_nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
prefix7606050378128311705at_nat: ( produc7491599851749785783at_nat > option_list_o ) > $o ).
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prefix4146821613196974222at_nat: ( produc7312547201344536560at_nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__Set__Oset_It__Nat__Onat_J,type,
prefix9179566592845446436et_nat: ( set_nat > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__comp_001_Eo,type,
prefix454693708527911765comp_o: option_list_o > option_list_o > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
collec4355076819549272527list_o: ( option_list_o > $o ) > set_option_list_o ).
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_Oimage_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
image_4309273772856505399_ereal: ( nat > extended_ereal ) > set_nat > set_Extended_ereal ).
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__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
image_4575287668734308173list_o: ( nat > option_list_o ) > set_nat > set_option_list_o ).
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__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
image_5401538521246155887_o_nat: ( option_list_o > nat ) > set_option_list_o > set_nat ).
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__Interval_Oord__class_OlessThan_001t__Extended____Real__Oereal,type,
set_or4817620380262991601_ereal: extended_ereal > set_Extended_ereal ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
set_or6815529063353638119list_o: set_option_list_o > set_se4724700259957787812list_o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
member_option_list_o: option_list_o > set_option_list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
member5519481007471526743_ereal: set_Extended_ereal > set_se6634062954251873166_ereal > $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__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
member7936360586209771373list_o: set_option_list_o > set_se4724700259957787812list_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
member3860231779568403053on_nat: set_option_nat > set_set_option_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1869216328726507724at_nat: set_Sum_sum_nat_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_v_l,type,
l: nat ).
% Relevant facts (1267)
thf(fact_0_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_1_dependent__encoding,axiom,
! [E1: set_nat > option_list_o,E2: set_nat > nat > option_list_o] :
( ( prefix9179566592845446436et_nat @ E1 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( dom_set_nat_list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix7606050378128311705at_nat @ ( prefix3505681232979180120at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_2_dependent__encoding,axiom,
! [E1: option_list_o > option_list_o,E2: option_list_o > nat > option_list_o] :
( ( prefix6898455076116784582list_o @ E1 )
=> ( ! [X2: option_list_o] :
( ( member_option_list_o @ X2 @ ( dom_op3781509917836105484list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix6912799275947769473_o_nat @ ( prefix5571864788691152052_o_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_3_dependent__encoding,axiom,
! [E1: set_nat > option_list_o,E2: set_nat > product_prod_nat_nat > option_list_o] :
( ( prefix9179566592845446436et_nat @ E1 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( dom_set_nat_list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix4146821613196974222at_nat @ ( prefix1222217955897696377at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_4_dependent__encoding,axiom,
! [E1: option_list_o > option_list_o,E2: option_list_o > product_prod_nat_nat > option_list_o] :
( ( prefix6898455076116784582list_o @ E1 )
=> ( ! [X2: option_list_o] :
( ( member_option_list_o @ X2 @ ( dom_op3781509917836105484list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix3213284823089634726at_nat @ ( prefix5045777197273762589at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_5_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > product_prod_nat_nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix726093921677949892at_nat @ ( prefix5760520046686399407at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_6_dependent__encoding,axiom,
! [E1: product_prod_nat_nat > option_list_o,E2: product_prod_nat_nat > nat > option_list_o] :
( ( prefix5049516368958873059at_nat @ E1 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( dom_Pr5202128127435093295list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix1851580905645250974at_nat @ ( prefix2400472653643684305at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_7_dependent__encoding,axiom,
! [E1: product_prod_nat_nat > option_list_o,E2: product_prod_nat_nat > product_prod_nat_nat > option_list_o] :
( ( prefix5049516368958873059at_nat @ E1 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( dom_Pr5202128127435093295list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix8803165839153646153at_nat @ ( prefix8375143551930000832at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_8_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix5049516368958873059at_nat @ ( prefix1356340002065705634at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_9_encoding__imp__inj,axiom,
! [F: product_prod_nat_nat > option_list_o] :
( ( prefix5049516368958873059at_nat @ F )
=> ( inj_on8168085447206903444list_o @ F @ ( dom_Pr5202128127435093295list_o @ F ) ) ) ).
% encoding_imp_inj
thf(fact_10_encoding__imp__inj,axiom,
! [F: nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ F )
=> ( inj_on1630180835328728801list_o @ F @ ( dom_nat_list_o @ F ) ) ) ).
% encoding_imp_inj
thf(fact_11_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X3: option_list_o,Y2: option_list_o] : ( prefix454693708527911765comp_o @ Y2 @ X3 ) ) ) ).
% opt_comp_sym
thf(fact_12_map__le__def,axiom,
( map_le_nat_list_o
= ( ^ [M_1: nat > option_list_o,M_2: nat > option_list_o] :
! [X3: nat] :
( ( member_nat @ X3 @ ( dom_nat_list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_13_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_14_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_15_lessThan__iff,axiom,
! [I: set_option_list_o,K: set_option_list_o] :
( ( member7936360586209771373list_o @ I @ ( set_or6815529063353638119list_o @ K ) )
= ( ord_le4476516537835661936list_o @ I @ K ) ) ).
% lessThan_iff
thf(fact_16_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_17_map__add__dom__app__simps_I3_J,axiom,
! [M: nat,L2: nat > option_list_o,L1: nat > option_list_o] :
( ~ ( member_nat @ M @ ( dom_nat_list_o @ L2 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_18_map__add__dom__app__simps_I2_J,axiom,
! [M: nat,L1: nat > option_list_o,L2: nat > option_list_o] :
( ~ ( member_nat @ M @ ( dom_nat_list_o @ L1 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_19_map__add__dom__app__simps_I1_J,axiom,
! [M: nat,L2: nat > option_list_o,L1: nat > option_list_o] :
( ( member_nat @ M @ ( dom_nat_list_o @ L2 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_20_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_21_Sup__lessThan,axiom,
! [Y: extended_ereal] :
( ( comple8415311339701865915_ereal @ ( set_or4817620380262991601_ereal @ Y ) )
= Y ) ).
% Sup_lessThan
thf(fact_22_map__add__assoc,axiom,
! [M1: nat > option_list_o,M2: nat > option_list_o,M3: nat > option_list_o] :
( ( map_add_nat_list_o @ M1 @ ( map_add_nat_list_o @ M2 @ M3 ) )
= ( map_add_nat_list_o @ ( map_add_nat_list_o @ M1 @ M2 ) @ M3 ) ) ).
% map_add_assoc
thf(fact_23_inj__on__map__add__dom,axiom,
! [M: nat > option_list_o,M4: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ ( map_add_nat_list_o @ M @ M4 ) @ ( dom_nat_list_o @ M4 ) )
= ( inj_on1630180835328728801list_o @ M4 @ ( dom_nat_list_o @ M4 ) ) ) ).
% inj_on_map_add_dom
thf(fact_24_map__le__refl,axiom,
! [F: nat > option_list_o] : ( map_le_nat_list_o @ F @ F ) ).
% map_le_refl
thf(fact_25_map__le__trans,axiom,
! [M1: nat > option_list_o,M2: nat > option_list_o,M3: nat > option_list_o] :
( ( map_le_nat_list_o @ M1 @ M2 )
=> ( ( map_le_nat_list_o @ M2 @ M3 )
=> ( map_le_nat_list_o @ M1 @ M3 ) ) ) ).
% map_le_trans
thf(fact_26_map__le__antisym,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ G )
=> ( ( map_le_nat_list_o @ G @ F )
=> ( F = G ) ) ) ).
% map_le_antisym
thf(fact_27_map__le__map__add,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] : ( map_le_nat_list_o @ F @ ( map_add_nat_list_o @ G @ F ) ) ).
% map_le_map_add
thf(fact_28_map__add__le__mapE,axiom,
! [F: nat > option_list_o,G: nat > option_list_o,H: nat > option_list_o] :
( ( map_le_nat_list_o @ ( map_add_nat_list_o @ F @ G ) @ H )
=> ( map_le_nat_list_o @ G @ H ) ) ).
% map_add_le_mapE
thf(fact_29_map__add__le__mapI,axiom,
! [F: nat > option_list_o,H: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ H )
=> ( ( map_le_nat_list_o @ G @ H )
=> ( map_le_nat_list_o @ ( map_add_nat_list_o @ F @ G ) @ H ) ) ) ).
% map_add_le_mapI
thf(fact_30_map__add__subsumed1,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ G )
=> ( ( map_add_nat_list_o @ F @ G )
= G ) ) ).
% map_add_subsumed1
thf(fact_31_map__add__subsumed2,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ G )
=> ( ( map_add_nat_list_o @ G @ F )
= G ) ) ).
% map_add_subsumed2
thf(fact_32_map__le__iff__map__add__commute,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ ( map_add_nat_list_o @ F @ G ) )
= ( ( map_add_nat_list_o @ F @ G )
= ( map_add_nat_list_o @ G @ F ) ) ) ).
% map_le_iff_map_add_commute
thf(fact_33_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_34_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_35_lessThan__strict__subset__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_36_is__encodingD,axiom,
! [E: product_prod_nat_nat > option_list_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( prefix5049516368958873059at_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_37_is__encodingD,axiom,
! [E: nat > option_list_o,X: nat,Y: nat] :
( ( prefix3558185134189398382ng_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_38_is__encodingI__2,axiom,
! [E: product_prod_nat_nat > option_list_o] :
( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( prefix5049516368958873059at_nat @ E ) ) ).
% is_encodingI_2
thf(fact_39_is__encodingI__2,axiom,
! [E: nat > option_list_o] :
( ! [X2: nat,Y3: nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( prefix3558185134189398382ng_nat @ E ) ) ).
% is_encodingI_2
thf(fact_40_finite__imp__Sup__less,axiom,
! [X4: set_Extended_ereal,X: extended_ereal,A: extended_ereal] :
( ( finite7198162374296863863_ereal @ X4 )
=> ( ( member2350847679896131959_ereal @ X @ X4 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ X4 )
=> ( ord_le1188267648640031866_ereal @ X2 @ A ) )
=> ( ord_le1188267648640031866_ereal @ ( comple8415311339701865915_ereal @ X4 ) @ A ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_41_finite__imp__Sup__less,axiom,
! [X4: set_nat,X: nat,A: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat @ X @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_nat @ X2 @ A ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_42_finite__Union,axiom,
! [A2: set_se4724700259957787812list_o] :
( ( finite4680056123773102701list_o @ A2 )
=> ( ! [M6: set_option_list_o] :
( ( member7936360586209771373list_o @ M6 @ A2 )
=> ( finite7007496012504252301list_o @ M6 ) )
=> ( finite7007496012504252301list_o @ ( comple1085286230969029553list_o @ A2 ) ) ) ) ).
% finite_Union
thf(fact_43_finite__Union,axiom,
! [A2: set_set_option_nat] :
( ( finite1464753433994532717on_nat @ A2 )
=> ( ! [M6: set_option_nat] :
( ( member3860231779568403053on_nat @ M6 @ A2 )
=> ( finite5523153139673422903on_nat @ M6 ) )
=> ( finite5523153139673422903on_nat @ ( comple3326054718015411497on_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_44_finite__Union,axiom,
! [A2: set_se3873067930692246379at_nat] :
( ( finite5325900196762371532at_nat @ A2 )
=> ( ! [M6: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ M6 @ A2 )
=> ( finite6187706683773761046at_nat @ M6 ) )
=> ( finite6187706683773761046at_nat @ ( comple2155544827851854728at_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_45_finite__Union,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [M6: set_set_nat] :
( ( member_set_set_nat @ M6 @ A2 )
=> ( finite1152437895449049373et_nat @ M6 ) )
=> ( finite1152437895449049373et_nat @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_46_finite__Union,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ! [M6: set_nat_nat] :
( ( member_set_nat_nat @ M6 @ A2 )
=> ( finite2115694454571419734at_nat @ M6 ) )
=> ( finite2115694454571419734at_nat @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_47_finite__Union,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [M6: set_nat] :
( ( member_set_nat @ M6 @ A2 )
=> ( finite_finite_nat @ M6 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_48_psubset__card__mono,axiom,
! [B: set_option_nat,A2: set_option_nat] :
( ( finite5523153139673422903on_nat @ B )
=> ( ( ord_le1792839605950587050on_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite3630362424238721784on_nat @ A2 ) @ ( finite3630362424238721784on_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_49_psubset__card__mono,axiom,
! [B: set_Sum_sum_nat_nat,A2: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ B )
=> ( ( ord_le2904074325318523657at_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite8494011213269508311at_nat @ A2 ) @ ( finite8494011213269508311at_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_50_psubset__card__mono,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_less_set_set_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_51_psubset__card__mono,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_less_set_nat_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_52_psubset__card__mono,axiom,
! [B: set_option_list_o,A2: set_option_list_o] :
( ( finite7007496012504252301list_o @ B )
=> ( ( ord_le4476516537835661936list_o @ A2 @ B )
=> ( ord_less_nat @ ( finite3362998479529755404list_o @ A2 ) @ ( finite3362998479529755404list_o @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_53_psubset__card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_54_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I2: nat,J: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( member_nat @ J @ A2 )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( F @ I2 )
!= ( F @ J ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_55_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > option_list_o] :
( ! [I2: nat,J: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( member_nat @ J @ A2 )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( F @ I2 )
!= ( F @ J ) ) ) ) )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_56_less__Sup__iff,axiom,
! [A: extended_ereal,S: set_Extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( comple8415311339701865915_ereal @ S ) )
= ( ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ S )
& ( ord_le1188267648640031866_ereal @ A @ X3 ) ) ) ) ).
% less_Sup_iff
thf(fact_57_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M7: nat] :
( ( ord_less_nat @ K @ M7 )
=> ? [N4: nat] :
( ( ord_less_nat @ M7 @ N4 )
& ( member_nat @ N4 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_58_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_59_finite__graph__iff__finite__dom,axiom,
! [M: nat > option_list_o] :
( ( finite1686248346847189978list_o @ ( graph_nat_list_o @ M ) )
= ( finite_finite_nat @ ( dom_nat_list_o @ M ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_60_inj__onD,axiom,
! [F: set_nat > set_option_list_o,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on9154615605479958667list_o @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_61_inj__onD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_62_inj__onD,axiom,
! [F: option_list_o > nat,A2: set_option_list_o,X: option_list_o,Y: option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_option_list_o @ X @ A2 )
=> ( ( member_option_list_o @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_63_inj__onD,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_64_inj__onD,axiom,
! [F: nat > option_list_o,A2: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_65_inj__onI,axiom,
! [A2: set_set_nat,F: set_nat > set_option_list_o] :
( ! [X2: set_nat,Y3: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on9154615605479958667list_o @ F @ A2 ) ) ).
% inj_onI
thf(fact_66_inj__onI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ! [X2: set_nat,Y3: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on4604407203859583615et_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_67_inj__onI,axiom,
! [A2: set_option_list_o,F: option_list_o > nat] :
( ! [X2: option_list_o,Y3: option_list_o] :
( ( member_option_list_o @ X2 @ A2 )
=> ( ( member_option_list_o @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on2456431687840576515_o_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_68_inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_69_inj__onI,axiom,
! [A2: set_nat,F: nat > option_list_o] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ).
% inj_onI
thf(fact_70_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_71_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_72_UnionI,axiom,
! [X4: set_set_nat,C: set_set_set_nat,A2: set_nat] :
( ( member_set_set_nat @ X4 @ C )
=> ( ( member_set_nat @ A2 @ X4 )
=> ( member_set_nat @ A2 @ ( comple548664676211718543et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_73_UnionI,axiom,
! [X4: set_option_list_o,C: set_se4724700259957787812list_o,A2: option_list_o] :
( ( member7936360586209771373list_o @ X4 @ C )
=> ( ( member_option_list_o @ A2 @ X4 )
=> ( member_option_list_o @ A2 @ ( comple1085286230969029553list_o @ C ) ) ) ) ).
% UnionI
thf(fact_74_UnionI,axiom,
! [X4: set_nat,C: set_set_nat,A2: nat] :
( ( member_set_nat @ X4 @ C )
=> ( ( member_nat @ A2 @ X4 )
=> ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_75_Union__iff,axiom,
! [A2: set_nat,C: set_set_set_nat] :
( ( member_set_nat @ A2 @ ( comple548664676211718543et_nat @ C ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ C )
& ( member_set_nat @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_76_Union__iff,axiom,
! [A2: option_list_o,C: set_se4724700259957787812list_o] :
( ( member_option_list_o @ A2 @ ( comple1085286230969029553list_o @ C ) )
= ( ? [X3: set_option_list_o] :
( ( member7936360586209771373list_o @ X3 @ C )
& ( member_option_list_o @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_77_Union__iff,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
& ( member_nat @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_78_mem__Collect__eq,axiom,
! [A: option_list_o,P: option_list_o > $o] :
( ( member_option_list_o @ A @ ( collec4355076819549272527list_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_option_list_o] :
( ( collec4355076819549272527list_o
@ ^ [X3: option_list_o] : ( member_option_list_o @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_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_85_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_86_UnionE,axiom,
! [A2: set_nat,C: set_set_set_nat] :
( ( member_set_nat @ A2 @ ( comple548664676211718543et_nat @ C ) )
=> ~ ! [X5: set_set_nat] :
( ( member_set_nat @ A2 @ X5 )
=> ~ ( member_set_set_nat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_87_UnionE,axiom,
! [A2: option_list_o,C: set_se4724700259957787812list_o] :
( ( member_option_list_o @ A2 @ ( comple1085286230969029553list_o @ C ) )
=> ~ ! [X5: set_option_list_o] :
( ( member_option_list_o @ A2 @ X5 )
=> ~ ( member7936360586209771373list_o @ X5 @ C ) ) ) ).
% UnionE
thf(fact_88_UnionE,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X5: set_nat] :
( ( member_nat @ A2 @ X5 )
=> ~ ( member_set_nat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_89_finite__psubset__induct,axiom,
! [A2: set_option_nat,P: set_option_nat > $o] :
( ( finite5523153139673422903on_nat @ A2 )
=> ( ! [A3: set_option_nat] :
( ( finite5523153139673422903on_nat @ A3 )
=> ( ! [B2: set_option_nat] :
( ( ord_le1792839605950587050on_nat @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_90_finite__psubset__induct,axiom,
! [A2: set_Sum_sum_nat_nat,P: set_Sum_sum_nat_nat > $o] :
( ( finite6187706683773761046at_nat @ A2 )
=> ( ! [A3: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ A3 )
=> ( ! [B2: set_Sum_sum_nat_nat] :
( ( ord_le2904074325318523657at_nat @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_91_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A3: set_set_nat] :
( ( finite1152437895449049373et_nat @ A3 )
=> ( ! [B2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_92_finite__psubset__induct,axiom,
! [A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ! [A3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A3 )
=> ( ! [B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_93_finite__psubset__induct,axiom,
! [A2: set_option_list_o,P: set_option_list_o > $o] :
( ( finite7007496012504252301list_o @ A2 )
=> ( ! [A3: set_option_list_o] :
( ( finite7007496012504252301list_o @ A3 )
=> ( ! [B2: set_option_list_o] :
( ( ord_le4476516537835661936list_o @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_94_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [B2: set_nat] :
( ( ord_less_set_nat @ B2 @ A3 )
=> ( P @ B2 ) )
=> ( P @ A3 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_95_finite__UnionD,axiom,
! [A2: set_se4724700259957787812list_o] :
( ( finite7007496012504252301list_o @ ( comple1085286230969029553list_o @ A2 ) )
=> ( finite4680056123773102701list_o @ A2 ) ) ).
% finite_UnionD
thf(fact_96_finite__UnionD,axiom,
! [A2: set_set_option_nat] :
( ( finite5523153139673422903on_nat @ ( comple3326054718015411497on_nat @ A2 ) )
=> ( finite1464753433994532717on_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_97_finite__UnionD,axiom,
! [A2: set_se3873067930692246379at_nat] :
( ( finite6187706683773761046at_nat @ ( comple2155544827851854728at_nat @ A2 ) )
=> ( finite5325900196762371532at_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_98_finite__UnionD,axiom,
! [A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ ( comple548664676211718543et_nat @ A2 ) )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_99_finite__UnionD,axiom,
! [A2: set_set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( comple5448282615319421384at_nat @ A2 ) )
=> ( finite3586981331298542604at_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_100_finite__UnionD,axiom,
! [A2: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_101_inj__on__inverseI,axiom,
! [A2: set_set_nat,G: set_option_list_o > set_nat,F: set_nat > set_option_list_o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on9154615605479958667list_o @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_102_inj__on__inverseI,axiom,
! [A2: set_set_nat,G: set_nat > set_nat,F: set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on4604407203859583615et_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_103_inj__on__inverseI,axiom,
! [A2: set_option_list_o,G: nat > option_list_o,F: option_list_o > nat] :
( ! [X2: option_list_o] :
( ( member_option_list_o @ X2 @ A2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on2456431687840576515_o_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_104_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_105_inj__on__inverseI,axiom,
! [A2: set_nat,G: option_list_o > nat,F: nat > option_list_o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_106_inj__on__contraD,axiom,
! [F: set_nat > set_option_list_o,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on9154615605479958667list_o @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_107_inj__on__contraD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_108_inj__on__contraD,axiom,
! [F: option_list_o > nat,A2: set_option_list_o,X: option_list_o,Y: option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_option_list_o @ X @ A2 )
=> ( ( member_option_list_o @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_109_inj__on__contraD,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_110_inj__on__contraD,axiom,
! [F: nat > option_list_o,A2: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_111_inj__on__eq__iff,axiom,
! [F: set_nat > set_option_list_o,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on9154615605479958667list_o @ F @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_112_inj__on__eq__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_113_inj__on__eq__iff,axiom,
! [F: option_list_o > nat,A2: set_option_list_o,X: option_list_o,Y: option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F @ A2 )
=> ( ( member_option_list_o @ X @ A2 )
=> ( ( member_option_list_o @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_114_inj__on__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_115_inj__on__eq__iff,axiom,
! [F: nat > option_list_o,A2: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_116_inj__on__cong,axiom,
! [A2: set_set_nat,F: set_nat > set_option_list_o,G: set_nat > set_option_list_o] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on9154615605479958667list_o @ F @ A2 )
= ( inj_on9154615605479958667list_o @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_117_inj__on__cong,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on4604407203859583615et_nat @ F @ A2 )
= ( inj_on4604407203859583615et_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_118_inj__on__cong,axiom,
! [A2: set_option_list_o,F: option_list_o > nat,G: option_list_o > nat] :
( ! [A4: option_list_o] :
( ( member_option_list_o @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on2456431687840576515_o_nat @ F @ A2 )
= ( inj_on2456431687840576515_o_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_119_inj__on__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_120_inj__on__cong,axiom,
! [A2: set_nat,F: nat > option_list_o,G: nat > option_list_o] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on1630180835328728801list_o @ F @ A2 )
= ( inj_on1630180835328728801list_o @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_121_inj__on__def,axiom,
( inj_on9154615605479958667list_o
= ( ^ [F2: set_nat > set_option_list_o,A5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_122_inj__on__def,axiom,
( inj_on4604407203859583615et_nat
= ( ^ [F2: set_nat > set_nat,A5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_123_inj__on__def,axiom,
( inj_on2456431687840576515_o_nat
= ( ^ [F2: option_list_o > nat,A5: set_option_list_o] :
! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ A5 )
=> ! [Y2: option_list_o] :
( ( member_option_list_o @ Y2 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_124_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_125_inj__on__def,axiom,
( inj_on1630180835328728801list_o
= ( ^ [F2: nat > option_list_o,A5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_126_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_127_finite__enumerate__mono,axiom,
! [M: nat,N2: nat,S: set_nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_128_enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_129_card__psubset,axiom,
! [B: set_option_nat,A2: set_option_nat] :
( ( finite5523153139673422903on_nat @ B )
=> ( ( ord_le6937355464348597430on_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite3630362424238721784on_nat @ A2 ) @ ( finite3630362424238721784on_nat @ B ) )
=> ( ord_le1792839605950587050on_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_130_card__psubset,axiom,
! [B: set_Sum_sum_nat_nat,A2: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ B )
=> ( ( ord_le5967974642961909525at_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite8494011213269508311at_nat @ A2 ) @ ( finite8494011213269508311at_nat @ B ) )
=> ( ord_le2904074325318523657at_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_131_card__psubset,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) )
=> ( ord_less_set_nat_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_132_card__psubset,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) )
=> ( ord_less_set_set_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_133_card__psubset,axiom,
! [B: set_option_list_o,A2: set_option_list_o] :
( ( finite7007496012504252301list_o @ B )
=> ( ( ord_le1162937763994921316list_o @ A2 @ B )
=> ( ( ord_less_nat @ ( finite3362998479529755404list_o @ A2 ) @ ( finite3362998479529755404list_o @ B ) )
=> ( ord_le4476516537835661936list_o @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_134_card__psubset,axiom,
! [B: set_Extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( ord_less_nat @ ( finite6323036508664497142_ereal @ A2 ) @ ( finite6323036508664497142_ereal @ B ) )
=> ( ord_le5321083090456148570_ereal @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_135_card__psubset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_136_finite__enumerate,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [R: nat > nat] :
( ( monotone_on_nat_nat @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) @ ord_less_nat @ ord_less_nat @ R )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( R @ N4 ) @ S ) ) ) ) ).
% finite_enumerate
thf(fact_137_finite__enumerate__in__set,axiom,
! [S: set_nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_138_finite__enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N6: nat] :
( ( ord_less_nat @ N6 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N6 )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_139_finite__enum__ext,axiom,
! [X4: set_nat,Y4: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X4 ) )
=> ( ( infini8530281810654367211te_nat @ X4 @ I2 )
= ( infini8530281810654367211te_nat @ Y4 @ I2 ) ) )
=> ( ( finite_finite_nat @ X4 )
=> ( ( finite_finite_nat @ Y4 )
=> ( ( ( finite_card_nat @ X4 )
= ( finite_card_nat @ Y4 ) )
=> ( X4 = Y4 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_140_enumerate__mono,axiom,
! [M: nat,N2: nat,S: set_nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_141_psubsetD,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_142_psubsetD,axiom,
! [A2: set_option_list_o,B: set_option_list_o,C2: option_list_o] :
( ( ord_le4476516537835661936list_o @ A2 @ B )
=> ( ( member_option_list_o @ C2 @ A2 )
=> ( member_option_list_o @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_143_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_144_psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% psubset_trans
thf(fact_145_psubset__trans,axiom,
! [A2: set_option_list_o,B: set_option_list_o,C: set_option_list_o] :
( ( ord_le4476516537835661936list_o @ A2 @ B )
=> ( ( ord_le4476516537835661936list_o @ B @ C )
=> ( ord_le4476516537835661936list_o @ A2 @ C ) ) ) ).
% psubset_trans
thf(fact_146_subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_147_subsetI,axiom,
! [A2: set_option_list_o,B: set_option_list_o] :
( ! [X2: option_list_o] :
( ( member_option_list_o @ X2 @ A2 )
=> ( member_option_list_o @ X2 @ B ) )
=> ( ord_le1162937763994921316list_o @ A2 @ B ) ) ).
% subsetI
thf(fact_148_subsetI,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ X2 @ B ) )
=> ( ord_le1644982726543182158_ereal @ A2 @ B ) ) ).
% subsetI
thf(fact_149_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_150_subset__antisym,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_151_subset__antisym,axiom,
! [A2: set_option_list_o,B: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ A2 @ B )
=> ( ( ord_le1162937763994921316list_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_152_subset__antisym,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_153_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_154_psubsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_155_psubsetI,axiom,
! [A2: set_option_list_o,B: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le4476516537835661936list_o @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_156_psubsetI,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le5321083090456148570_ereal @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_157_psubsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_158_lessThan__subset__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( set_or4817620380262991601_ereal @ X ) @ ( set_or4817620380262991601_ereal @ Y ) )
= ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_159_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_160_enumerate__mono__le__iff,axiom,
! [S: set_nat,M: nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_161_monotone__on__subset,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: set_nat > set_nat > $o,F: nat > set_nat,B: set_nat] :
( ( monoto6489329683466618047et_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto6489329683466618047et_nat @ B @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_162_monotone__on__subset,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: nat > nat > $o,F: set_nat > nat,B: set_set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto2923694778811248831at_nat @ B @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_163_monotone__on__subset,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: set_nat > set_nat > $o,F: set_nat > set_nat,B: set_set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto1748750089227133045et_nat @ B @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_164_monotone__on__subset,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monotone_on_nat_nat @ B @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_165_monotone__on__def,axiom,
( monoto6489329683466618047et_nat
= ( ^ [A5: set_nat,Orda2: nat > nat > $o,Ordb2: set_nat > set_nat > $o,F2: nat > set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_166_monotone__on__def,axiom,
( monoto2923694778811248831at_nat
= ( ^ [A5: set_set_nat,Orda2: set_nat > set_nat > $o,Ordb2: nat > nat > $o,F2: set_nat > nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A5 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_167_monotone__on__def,axiom,
( monoto1748750089227133045et_nat
= ( ^ [A5: set_set_nat,Orda2: set_nat > set_nat > $o,Ordb2: set_nat > set_nat > $o,F2: set_nat > set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A5 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_168_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A5: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F2: nat > nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_169_monotone__onI,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: set_nat > set_nat > $o,F: nat > set_nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto6489329683466618047et_nat @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_170_monotone__onI,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: nat > nat > $o,F: set_nat > nat] :
( ! [X2: set_nat,Y3: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y3 @ A2 )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto2923694778811248831at_nat @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_171_monotone__onI,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: set_nat > set_nat > $o,F: set_nat > set_nat] :
( ! [X2: set_nat,Y3: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y3 @ A2 )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto1748750089227133045et_nat @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_172_monotone__onI,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_173_monotone__onD,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: set_nat > set_nat > $o,F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_174_monotone__onD,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: nat > nat > $o,F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_175_monotone__onD,axiom,
! [A2: set_set_nat,Orda: set_nat > set_nat > $o,Ordb: set_nat > set_nat > $o,F: set_nat > set_nat,X: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_176_monotone__onD,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_177_mono__on__subset,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monotone_on_nat_nat @ B @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_178_mono__on__subset,axiom,
! [A2: set_set_nat,F: set_nat > nat,B: set_set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto2923694778811248831at_nat @ B @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_179_mono__on__subset,axiom,
! [A2: set_nat,F: nat > set_nat,B: set_nat] :
( ( monoto6489329683466618047et_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto6489329683466618047et_nat @ B @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_180_mono__on__subset,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B: set_set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto1748750089227133045et_nat @ B @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_181_mono__on__subset,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > nat,B: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ A2 @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto2580034644210098551al_nat @ B @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_182_mono__on__subset,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > set_nat,B: set_Extended_ereal] :
( ( monoto350475336054780205et_nat @ A2 @ ord_le1083603963089353582_ereal @ ord_less_eq_set_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto350475336054780205et_nat @ B @ ord_le1083603963089353582_ereal @ ord_less_eq_set_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_183_mono__on__subset,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > set_Extended_ereal,B: set_Extended_ereal] :
( ( monoto1076656197419758151_ereal @ A2 @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto1076656197419758151_ereal @ B @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F ) ) ) ).
% mono_on_subset
thf(fact_184_mono__on__subset,axiom,
! [A2: set_nat,F: nat > set_Extended_ereal,B: set_nat] :
( ( monoto6788471982328799797_ereal @ A2 @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto6788471982328799797_ereal @ B @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F ) ) ) ).
% mono_on_subset
thf(fact_185_mono__on__subset,axiom,
! [A2: set_se6634062954251873166_ereal,F: set_Extended_ereal > nat,B: set_se6634062954251873166_ereal] :
( ( monoto375287072342888279al_nat @ A2 @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F )
=> ( ( ord_le5287700718633833262_ereal @ B @ A2 )
=> ( monoto375287072342888279al_nat @ B @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_186_mono__on__subset,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > set_set_nat,B: set_Extended_ereal] :
( ( monoto6111297408403839459et_nat @ A2 @ ord_le1083603963089353582_ereal @ ord_le6893508408891458716et_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto6111297408403839459et_nat @ B @ ord_le1083603963089353582_ereal @ ord_le6893508408891458716et_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_187_mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_188_mono__onI,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( ord_less_eq_set_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2923694778811248831at_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_189_mono__onI,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_set_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6489329683466618047et_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_onI
thf(fact_190_mono__onI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( ord_less_eq_set_nat @ R @ S3 )
=> ( ord_less_eq_set_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1748750089227133045et_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_onI
thf(fact_191_mono__onI,axiom,
! [A2: set_nat,F: nat > set_Extended_ereal] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6788471982328799797_ereal @ A2 @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F ) ) ).
% mono_onI
thf(fact_192_mono__onI,axiom,
! [A2: set_se6634062954251873166_ereal,F: set_Extended_ereal > nat] :
( ! [R: set_Extended_ereal,S3: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ R @ A2 )
=> ( ( member5519481007471526743_ereal @ S3 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto375287072342888279al_nat @ A2 @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_193_mono__onI,axiom,
! [A2: set_set_nat,F: set_nat > set_Extended_ereal] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( ord_less_eq_set_nat @ R @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto3364847110614814975_ereal @ A2 @ ord_less_eq_set_nat @ ord_le1644982726543182158_ereal @ F ) ) ).
% mono_onI
thf(fact_194_mono__onI,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto5249346468214351221et_nat @ A2 @ ord_less_eq_nat @ ord_le6893508408891458716et_nat @ F ) ) ).
% mono_onI
thf(fact_195_mono__onI,axiom,
! [A2: set_nat,F: nat > $o > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_o_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto5986239079323330272_o_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_o_nat @ F ) ) ).
% mono_onI
thf(fact_196_mono__onI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ! [R: set_set_nat,S3: set_set_nat] :
( ( member_set_set_nat @ R @ A2 )
=> ( ( member_set_set_nat @ S3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4510150628252253557at_nat @ A2 @ ord_le6893508408891458716et_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_197_mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_198_mono__onD,axiom,
! [A2: set_set_nat,F: set_nat > nat,R2: set_nat,S2: set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_199_mono__onD,axiom,
! [A2: set_nat,F: nat > set_nat,R2: nat,S2: nat] :
( ( monoto6489329683466618047et_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_200_mono__onD,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,R2: set_nat,S2: set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_201_mono__onD,axiom,
! [A2: set_nat,F: nat > set_Extended_ereal,R2: nat,S2: nat] :
( ( monoto6788471982328799797_ereal @ A2 @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_202_mono__onD,axiom,
! [A2: set_se6634062954251873166_ereal,F: set_Extended_ereal > nat,R2: set_Extended_ereal,S2: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ A2 @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F )
=> ( ( member5519481007471526743_ereal @ R2 @ A2 )
=> ( ( member5519481007471526743_ereal @ S2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_203_mono__onD,axiom,
! [A2: set_set_nat,F: set_nat > set_Extended_ereal,R2: set_nat,S2: set_nat] :
( ( monoto3364847110614814975_ereal @ A2 @ ord_less_eq_set_nat @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_204_mono__onD,axiom,
! [A2: set_nat,F: nat > set_set_nat,R2: nat,S2: nat] :
( ( monoto5249346468214351221et_nat @ A2 @ ord_less_eq_nat @ ord_le6893508408891458716et_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_205_mono__onD,axiom,
! [A2: set_nat,F: nat > $o > nat,R2: nat,S2: nat] :
( ( monoto5986239079323330272_o_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_o_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_o_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_206_mono__onD,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat,R2: set_set_nat,S2: set_set_nat] :
( ( monoto4510150628252253557at_nat @ A2 @ ord_le6893508408891458716et_nat @ ord_less_eq_nat @ F )
=> ( ( member_set_set_nat @ R2 @ A2 )
=> ( ( member_set_set_nat @ S2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_207_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monotone_on_nat_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_208_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_nat,B: set_nat] :
( ( monoto6489329683466618047et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto6489329683466618047et_nat @ B @ Less_eq @ ord_less_eq_set_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_209_ord_Omono__on__subset,axiom,
! [A2: set_Extended_ereal,Less_eq: extended_ereal > extended_ereal > $o,F: extended_ereal > nat,B: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto2580034644210098551al_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_210_ord_Omono__on__subset,axiom,
! [A2: set_Extended_ereal,Less_eq: extended_ereal > extended_ereal > $o,F: extended_ereal > set_nat,B: set_Extended_ereal] :
( ( monoto350475336054780205et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto350475336054780205et_nat @ B @ Less_eq @ ord_less_eq_set_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_211_ord_Omono__on__subset,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > nat,B: set_set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto2923694778811248831at_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_212_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_Extended_ereal,B: set_nat] :
( ( monoto6788471982328799797_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto6788471982328799797_ereal @ B @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_213_ord_Omono__on__subset,axiom,
! [A2: set_Extended_ereal,Less_eq: extended_ereal > extended_ereal > $o,F: extended_ereal > set_Extended_ereal,B: set_Extended_ereal] :
( ( monoto1076656197419758151_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A2 )
=> ( monoto1076656197419758151_ereal @ B @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_214_ord_Omono__on__subset,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_nat,B: set_set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( monoto1748750089227133045et_nat @ B @ Less_eq @ ord_less_eq_set_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_215_ord_Omono__on__subset,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > nat,B: set_option_list_o] :
( ( monoto2661105143663773837_o_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_le1162937763994921316list_o @ B @ A2 )
=> ( monoto2661105143663773837_o_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_216_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_set_nat,B: set_nat] :
( ( monoto5249346468214351221et_nat @ A2 @ Less_eq @ ord_le6893508408891458716et_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( monoto5249346468214351221et_nat @ B @ Less_eq @ ord_le6893508408891458716et_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_217_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_218_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_219_ord_Omono__on__def,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > nat] :
( ( monoto2923694778811248831at_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: set_nat,S4: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_220_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_221_ord_Omono__on__def,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
= ( ! [R3: set_nat,S4: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_222_ord_Omono__on__def,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > nat] :
( ( monoto2661105143663773837_o_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: option_list_o,S4: option_list_o] :
( ( ( member_option_list_o @ R3 @ A2 )
& ( member_option_list_o @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_223_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_set_nat] :
( ( monoto5249346468214351221et_nat @ A2 @ Less_eq @ ord_le6893508408891458716et_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_224_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > $o > nat] :
( ( monoto5986239079323330272_o_nat @ A2 @ Less_eq @ ord_less_eq_o_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_o_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_225_ord_Omono__on__def,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_Extended_ereal] :
( ( monoto3364847110614814975_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
= ( ! [R3: set_nat,S4: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_226_ord_Omono__on__def,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > set_nat] :
( ( monoto3365026004806434883et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
= ( ! [R3: option_list_o,S4: option_list_o] :
( ( ( member_option_list_o @ R3 @ A2 )
& ( member_option_list_o @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_227_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_228_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_set_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6489329683466618047et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F ) ) ).
% ord.mono_onI
thf(fact_229_ord_Omono__onI,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > nat] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2923694778811248831at_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_230_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_Extended_ereal] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6788471982328799797_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ).
% ord.mono_onI
thf(fact_231_ord_Omono__onI,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_nat] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_set_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1748750089227133045et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F ) ) ).
% ord.mono_onI
thf(fact_232_ord_Omono__onI,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > nat] :
( ! [R: option_list_o,S3: option_list_o] :
( ( member_option_list_o @ R @ A2 )
=> ( ( member_option_list_o @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2661105143663773837_o_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_233_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_set_nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_le6893508408891458716et_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto5249346468214351221et_nat @ A2 @ Less_eq @ ord_le6893508408891458716et_nat @ F ) ) ).
% ord.mono_onI
thf(fact_234_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > $o > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_o_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto5986239079323330272_o_nat @ A2 @ Less_eq @ ord_less_eq_o_nat @ F ) ) ).
% ord.mono_onI
thf(fact_235_ord_Omono__onI,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_Extended_ereal] :
( ! [R: set_nat,S3: set_nat] :
( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto3364847110614814975_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ).
% ord.mono_onI
thf(fact_236_ord_Omono__onI,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > set_nat] :
( ! [R: option_list_o,S3: option_list_o] :
( ( member_option_list_o @ R @ A2 )
=> ( ( member_option_list_o @ S3 @ A2 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_set_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto3365026004806434883et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F ) ) ).
% ord.mono_onI
thf(fact_237_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_238_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_nat,R2: nat,S2: nat] :
( ( monoto6489329683466618047et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_239_ord_Omono__onD,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > nat,R2: set_nat,S2: set_nat] :
( ( monoto2923694778811248831at_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_240_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_Extended_ereal,R2: nat,S2: nat] :
( ( monoto6788471982328799797_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_241_ord_Omono__onD,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_nat,R2: set_nat,S2: set_nat] :
( ( monoto1748750089227133045et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_242_ord_Omono__onD,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > nat,R2: option_list_o,S2: option_list_o] :
( ( monoto2661105143663773837_o_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_option_list_o @ R2 @ A2 )
=> ( ( member_option_list_o @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_243_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_set_nat,R2: nat,S2: nat] :
( ( monoto5249346468214351221et_nat @ A2 @ Less_eq @ ord_le6893508408891458716et_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_244_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > $o > nat,R2: nat,S2: nat] :
( ( monoto5986239079323330272_o_nat @ A2 @ Less_eq @ ord_less_eq_o_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_o_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_245_ord_Omono__onD,axiom,
! [A2: set_set_nat,Less_eq: set_nat > set_nat > $o,F: set_nat > set_Extended_ereal,R2: set_nat,S2: set_nat] :
( ( monoto3364847110614814975_ereal @ A2 @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_246_ord_Omono__onD,axiom,
! [A2: set_option_list_o,Less_eq: option_list_o > option_list_o > $o,F: option_list_o > set_nat,R2: option_list_o,S2: option_list_o] :
( ( monoto3365026004806434883et_nat @ A2 @ Less_eq @ ord_less_eq_set_nat @ F )
=> ( ( member_option_list_o @ R2 @ A2 )
=> ( ( member_option_list_o @ S2 @ A2 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_247_in__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_248_in__mono,axiom,
! [A2: set_option_list_o,B: set_option_list_o,X: option_list_o] :
( ( ord_le1162937763994921316list_o @ A2 @ B )
=> ( ( member_option_list_o @ X @ A2 )
=> ( member_option_list_o @ X @ B ) ) ) ).
% in_mono
thf(fact_249_in__mono,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,X: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( member2350847679896131959_ereal @ X @ A2 )
=> ( member2350847679896131959_ereal @ X @ B ) ) ) ).
% in_mono
thf(fact_250_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_251_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_252_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_253_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_254_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_255_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_256_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_257_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_258_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_259_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_260_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A5: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_261_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_262_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ A2 @ ord_less_nat @ ord_less_set_nat @ F )
=> ( monoto6489329683466618047et_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_263_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_264_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ A2 @ ord_less_nat @ ord_less_set_nat @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_265_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_266_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_267_Sup__subset__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_268_Union__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_269_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A5 @ B3 )
| ( A5 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_270_subset__psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_271_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_272_psubset__subset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_273_psubset__imp__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_274_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B3 )
& ( A5 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_275_psubsetE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_276_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_277_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_278_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_279_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_280_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_281_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_282_complete__interval,axiom,
! [A: nat,B4: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B4 )
=> ( ( P @ A )
=> ( ~ ( P @ B4 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B4 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_283_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_284_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 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_285_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_286_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 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_287_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_288_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_289_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_290_Sup__eqI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A2 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [Y3: set_nat] :
( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ A2 )
=> ( ord_less_eq_set_nat @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_291_Sup__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B )
& ( ord_less_eq_set_nat @ A4 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_292_Sup__least,axiom,
! [A2: set_set_nat,Z3: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z3 ) ) ).
% Sup_least
thf(fact_293_Sup__upper,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_294_Sup__le__iff,axiom,
! [A2: set_set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B4 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B4 ) ) ) ) ).
% Sup_le_iff
thf(fact_295_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_296_cSup__eq__maximum,axiom,
! [Z3: set_nat,X4: set_set_nat] :
( ( member_set_nat @ Z3 @ X4 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_297_cSup__eq__maximum,axiom,
! [Z3: nat,X4: set_nat] :
( ( member_nat @ Z3 @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_298_inj__on__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_299_inj__on__subset,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( inj_on1630180835328728801list_o @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_300_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_301_subset__inj__on,axiom,
! [F: nat > option_list_o,B: set_nat,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_302_Union__subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
& ( ord_less_eq_set_nat @ X2 @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_303_Union__upper,axiom,
! [B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_304_Union__least,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ X5 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_305_enumerate__in__set,axiom,
! [S: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ S ) ) ).
% enumerate_in_set
thf(fact_306_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N6: nat] :
( ( infini8530281810654367211te_nat @ S @ N6 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_307_strict__mono__on__imp__inj__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_308_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_309_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > option_list_o] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ) )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_310_le__cSup__finite,axiom,
! [X4: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( ( member_set_nat @ X @ X4 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_311_le__cSup__finite,axiom,
! [X4: set_nat,X: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat @ X @ X4 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_312_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_313_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N2 )
& ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_314_finite__nat__bounded,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_315_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_316_map__le__implies__dom__le,axiom,
! [F: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F @ G )
=> ( ord_less_eq_set_nat @ ( dom_nat_list_o @ F ) @ ( dom_nat_list_o @ G ) ) ) ).
% map_le_implies_dom_le
thf(fact_317_finite__subset__Union,axiom,
! [A2: set_nat,B6: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B6 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B6 )
=> ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_318_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_319_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_320_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_321_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_322_minf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% minf(8)
thf(fact_323_minf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% minf(6)
thf(fact_324_pinf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% pinf(8)
thf(fact_325_pinf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% pinf(6)
thf(fact_326_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_327_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_328_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_329_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_330_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M8: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M8 ) )
=> ~ ! [M7: nat] :
( ( P @ M7 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M7 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_331_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_332_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_333_le__enumerate,axiom,
! [S: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ).
% le_enumerate
thf(fact_334_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A4 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B9: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B9 @ B )
=> ( ( R2 @ A1 @ B9 )
=> ( ( R2 @ A22 @ B9 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_335_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_336_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_337_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_338_ord__le__eq__subst,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_339_ord__le__eq__subst,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_340_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_341_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_342_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B4: set_nat,C2: set_nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_343_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B4: set_nat,C2: set_nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_344_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_345_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_346_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_347_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_348_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_349_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_350_order__subst2,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_351_order__subst2,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_352_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_353_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_354_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_355_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B4: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_356_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_357_order__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_358_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A7: set_nat,B10: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ B10 )
& ( ord_less_eq_set_nat @ B10 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_359_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A7: nat,B10: nat] :
( ( ord_less_eq_nat @ A7 @ B10 )
& ( ord_less_eq_nat @ B10 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_360_antisym,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_361_antisym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_362_dual__order_Otrans,axiom,
! [B4: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B4 )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_363_dual__order_Otrans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_364_dual__order_Oantisym,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_365_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_366_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A7: set_nat,B10: set_nat] :
( ( ord_less_eq_set_nat @ B10 @ A7 )
& ( ord_less_eq_set_nat @ A7 @ B10 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_367_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A7: nat,B10: nat] :
( ( ord_less_eq_nat @ B10 @ A7 )
& ( ord_less_eq_nat @ A7 @ B10 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_368_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A4: nat,B9: nat] :
( ( ord_less_eq_nat @ A4 @ B9 )
=> ( P @ A4 @ B9 ) )
=> ( ! [A4: nat,B9: nat] :
( ( P @ B9 @ A4 )
=> ( P @ A4 @ B9 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_369_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_370_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_371_order_Otrans,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_372_order_Otrans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_373_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_374_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_375_ord__le__eq__trans,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_376_ord__le__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_377_ord__eq__le__trans,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( A = B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_378_ord__eq__le__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_379_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_380_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_381_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_382_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_383_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_384_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_385_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_386_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_387_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_388_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_389_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_390_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_391_order__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_392_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_393_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_394_ord__less__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_395_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_396_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_397_order__less__asym_H,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order_less_asym'
thf(fact_398_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_399_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_400_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_401_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_402_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( A != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_403_dual__order_Ostrict__trans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_404_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_405_order_Ostrict__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_406_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A4: nat,B9: nat] :
( ( ord_less_nat @ A4 @ B9 )
=> ( P @ A4 @ B9 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B9: nat] :
( ( P @ B9 @ A4 )
=> ( P @ A4 @ B9 ) )
=> ( P @ A @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_407_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N5: nat] :
( ( P3 @ N5 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_408_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_409_dual__order_Oasym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ A @ B4 ) ) ).
% dual_order.asym
thf(fact_410_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_411_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_412_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_413_ord__less__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_414_ord__eq__less__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_415_order_Oasym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order.asym
thf(fact_416_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_417_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_418_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_419_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_420_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_421_pinf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(3)
thf(fact_422_pinf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(4)
thf(fact_423_pinf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T3 ) ) ).
% pinf(5)
thf(fact_424_pinf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T3 @ X6 ) ) ).
% pinf(7)
thf(fact_425_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_426_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_427_minf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T3 ) ) ).
% minf(3)
thf(fact_428_minf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T3 ) ) ).
% minf(4)
thf(fact_429_minf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T3 ) ) ).
% minf(5)
thf(fact_430_minf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T3 @ X6 ) ) ).
% minf(7)
thf(fact_431_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F4 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_432_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_433_exists__subset__between,axiom,
! [A2: set_nat,N2: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ C )
& ( ( finite_card_nat @ B5 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_434_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_435_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_436_finite__le__enumerate,axiom,
! [S: set_nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_437_finite__enum__subset,axiom,
! [X4: set_nat,Y4: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X4 ) )
=> ( ( infini8530281810654367211te_nat @ X4 @ I2 )
= ( infini8530281810654367211te_nat @ Y4 @ I2 ) ) )
=> ( ( finite_finite_nat @ X4 )
=> ( ( finite_finite_nat @ Y4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X4 ) @ ( finite_card_nat @ Y4 ) )
=> ( ord_less_eq_set_nat @ X4 @ Y4 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_438_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_439_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_440_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_441_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_442_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_443_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_444_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B4: set_nat,C2: set_nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_445_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B4: nat,C2: nat] :
( ( ord_less_set_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_446_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_447_order__le__less__subst2,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_set_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_448_order__le__less__subst2,axiom,
! [A: set_nat,B4: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_449_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_set_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_450_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_451_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B4: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_452_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_453_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_454_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_455_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_456_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_457_order__neq__le__trans,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_set_nat @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_458_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_459_order__le__neq__trans,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_nat @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_460_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_461_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_462_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_463_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_464_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_465_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_466_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_467_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_468_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_469_dual__order_Ostrict__implies__order,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B4 @ A )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_470_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_471_order_Ostrict__implies__order,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_472_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_473_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B10: set_nat,A7: set_nat] :
( ( ord_less_eq_set_nat @ B10 @ A7 )
& ~ ( ord_less_eq_set_nat @ A7 @ B10 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_474_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B10: nat,A7: nat] :
( ( ord_less_eq_nat @ B10 @ A7 )
& ~ ( ord_less_eq_nat @ A7 @ B10 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_475_dual__order_Ostrict__trans2,axiom,
! [B4: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B4 )
=> ( ord_less_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_476_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_477_dual__order_Ostrict__trans1,axiom,
! [B4: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_less_set_nat @ C2 @ B4 )
=> ( ord_less_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_478_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_479_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B10: set_nat,A7: set_nat] :
( ( ord_less_eq_set_nat @ B10 @ A7 )
& ( A7 != B10 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_480_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B10: nat,A7: nat] :
( ( ord_less_eq_nat @ B10 @ A7 )
& ( A7 != B10 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_481_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B10: set_nat,A7: set_nat] :
( ( ord_less_set_nat @ B10 @ A7 )
| ( A7 = B10 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_482_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B10: nat,A7: nat] :
( ( ord_less_nat @ B10 @ A7 )
| ( A7 = B10 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_483_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A7: set_nat,B10: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ B10 )
& ~ ( ord_less_eq_set_nat @ B10 @ A7 ) ) ) ) ).
% order.strict_iff_not
thf(fact_484_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A7: nat,B10: nat] :
( ( ord_less_eq_nat @ A7 @ B10 )
& ~ ( ord_less_eq_nat @ B10 @ A7 ) ) ) ) ).
% order.strict_iff_not
thf(fact_485_order_Ostrict__trans2,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_486_order_Ostrict__trans2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_487_order_Ostrict__trans1,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_488_order_Ostrict__trans1,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_489_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A7: set_nat,B10: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ B10 )
& ( A7 != B10 ) ) ) ) ).
% order.strict_iff_order
thf(fact_490_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A7: nat,B10: nat] :
( ( ord_less_eq_nat @ A7 @ B10 )
& ( A7 != B10 ) ) ) ) ).
% order.strict_iff_order
thf(fact_491_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B10: set_nat] :
( ( ord_less_set_nat @ A7 @ B10 )
| ( A7 = B10 ) ) ) ) ).
% order.order_iff_strict
thf(fact_492_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B10: nat] :
( ( ord_less_nat @ A7 @ B10 )
| ( A7 = B10 ) ) ) ) ).
% order.order_iff_strict
thf(fact_493_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_494_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_495_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_496_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_497_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_498_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_499_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_500_nless__le,axiom,
! [A: set_nat,B4: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B4 ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_501_nless__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_502_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_503_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_504_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_505_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_506_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_507_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_508_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_509_ex__bij__betw__strict__mono__card,axiom,
! [M8: set_nat] :
( ( finite_finite_nat @ M8 )
=> ~ ! [H2: nat > nat] :
( ( bij_betw_nat_nat @ H2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ M8 ) ) @ M8 )
=> ~ ( monotone_on_nat_nat @ ( set_ord_lessThan_nat @ ( finite_card_nat @ M8 ) ) @ ord_less_nat @ ord_less_nat @ H2 ) ) ) ).
% ex_bij_betw_strict_mono_card
thf(fact_510_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat,B3: set_nat] :
? [G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ ( F2 @ X3 ) @ B3 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A5 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_511_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_512_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat,A8: set_nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A2 @ A8 )
= ( bij_betw_nat_nat @ G @ A2 @ A8 ) ) ) ).
% bij_betw_cong
thf(fact_513_bij__betw__ball,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat,Phi: nat > $o] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( Phi @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( Phi @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_514_bij__betw__inv,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ? [G4: nat > nat] : ( bij_betw_nat_nat @ G4 @ B @ A2 ) ) ).
% bij_betw_inv
thf(fact_515_bij__betwE,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_nat @ ( F @ X6 ) @ B ) ) ) ).
% bij_betwE
thf(fact_516_bij__betw__finite,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_517_bij__betw__imp__inj__on,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o] :
( ( bij_be1867499782348650780list_o @ F @ A2 @ B )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_518_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_519_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) ) ) ).
% bij_betw_same_card
thf(fact_520_bij__betw__iff__card,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F2: nat > nat] : ( bij_betw_nat_nat @ F2 @ A2 @ B ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_521_finite__same__card__bij,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ? [H2: nat > nat] : ( bij_betw_nat_nat @ H2 @ A2 @ B ) ) ) ) ).
% finite_same_card_bij
thf(fact_522_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_523_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N6: nat] :
( ~ ( P @ N6 )
=> ? [M9: nat] :
( ( ord_less_nat @ M9 @ N6 )
& ~ ( P @ M9 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_524_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N6: nat] :
( ! [M9: nat] :
( ( ord_less_nat @ M9 @ N6 )
=> ( P @ M9 ) )
=> ( P @ N6 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_525_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_526_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_527_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_528_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_529_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_530_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_531_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_532_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_533_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_534_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_535_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_536_finite__bij__enumerate,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) @ S ) ) ).
% finite_bij_enumerate
thf(fact_537_strict__mono__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).
% strict_mono_enumerate
thf(fact_538_finite__enumerate__step,axiom,
! [S: set_nat,N2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N2 ) @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N2 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_539_finite__enumerate__initial__segment,axiom,
! [S: set_nat,N2: nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) ) )
=> ( ( infini8530281810654367211te_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) @ N2 )
= ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_540_card__le__inj,axiom,
! [A2: set_nat,B: set_option_list_o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite7007496012504252301list_o @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite3362998479529755404list_o @ B ) )
=> ? [F5: nat > option_list_o] :
( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F5 @ A2 ) @ B )
& ( inj_on1630180835328728801list_o @ F5 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_541_card__le__inj,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ? [F5: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A2 ) @ B )
& ( inj_on_nat_nat @ F5 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_542_card__inj__on__le,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ B )
=> ( ( finite7007496012504252301list_o @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite3362998479529755404list_o @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_543_card__inj__on__le,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_544_inj__on__iff__card__le,axiom,
! [A2: set_nat,B: set_option_list_o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite7007496012504252301list_o @ B )
=> ( ( ? [F2: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A2 )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A2 ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite3362998479529755404list_o @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_545_inj__on__iff__card__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_546_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_547_image__eqI,axiom,
! [B4: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B4
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_548_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_549_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_550_bij__betw__Suc,axiom,
! [M8: set_nat,N: set_nat] :
( ( bij_betw_nat_nat @ suc @ M8 @ N )
= ( ( image_nat_nat @ suc @ M8 )
= N ) ) ).
% bij_betw_Suc
thf(fact_551_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_552_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_553_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_554_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_555_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_556_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_557_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_558_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_559_Int__UNIV,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_560_finite__Int,axiom,
! [F4: set_nat,G5: set_nat] :
( ( ( finite_finite_nat @ F4 )
| ( finite_finite_nat @ G5 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G5 ) ) ) ).
% finite_Int
thf(fact_561_Int__subset__iff,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_562_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_563_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_564_image__Int__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_565_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_566_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_567_bij__betw__imp__surj,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_568_bij__is__surj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_569_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_570_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_571_IntE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_572_IntD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_573_IntD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_574_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_575_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_576_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_577_image__iff,axiom,
! [Z3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_578_range__eqI,axiom,
! [B4: nat,F: nat > nat,X: nat] :
( ( B4
= ( F @ X ) )
=> ( member_nat @ B4 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_579_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_580_image__cong,axiom,
! [M8: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M8 = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M8 )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_581_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_582_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_583_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_584_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B4: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_585_Int__UNIV__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_586_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_587_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_588_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_589_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_590_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_591_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_592_image__Int,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% image_Int
thf(fact_593_image__Int,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( image_4575287668734308173list_o @ F @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_in6922378751903173298list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ ( image_4575287668734308173list_o @ F @ B ) ) ) ) ).
% image_Int
thf(fact_594_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_595_inj__image__mem__iff,axiom,
! [F: nat > option_list_o,A: nat,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( member_option_list_o @ ( F @ A ) @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_596_inj__image__mem__iff,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_597_inj__image__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_image_eq_iff
thf(fact_598_inj__image__eq__iff,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( ( image_4575287668734308173list_o @ F @ A2 )
= ( image_4575287668734308173list_o @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_image_eq_iff
thf(fact_599_range__ex1__eq,axiom,
! [F: nat > option_list_o,B4: option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( member_option_list_o @ B4 @ ( image_4575287668734308173list_o @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B4
= ( F @ X3 ) )
& ! [Y2: nat] :
( ( B4
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_600_range__ex1__eq,axiom,
! [F: nat > nat,B4: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B4
= ( F @ X3 ) )
& ! [Y2: nat] :
( ( B4
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_601_range__inj__infinite,axiom,
! [F: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ~ ( finite7007496012504252301list_o @ ( image_4575287668734308173list_o @ F @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_602_range__inj__infinite,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ~ ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_603_Sup__inter__less__eq,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_604_inj__on__image__Int,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_605_inj__on__image__Int,axiom,
! [F: nat > option_list_o,C: set_nat,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( image_4575287668734308173list_o @ F @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_in6922378751903173298list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ ( image_4575287668734308173list_o @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_606_Union__Int__subset,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_607_mono__inf,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_608_mono__inf,axiom,
! [F: set_nat > nat,A2: set_nat,B: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_609_mono__inf,axiom,
! [F: nat > set_nat,A2: nat,B: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( F @ ( inf_inf_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_610_mono__inf,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A2 @ B ) ) @ ( inf_inf_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_611_infinite__iff__countable__subset,axiom,
! [S: set_option_list_o] :
( ( ~ ( finite7007496012504252301list_o @ S ) )
= ( ? [F2: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_612_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_613_infinite__countable__subset,axiom,
! [S: set_option_list_o] :
( ~ ( finite7007496012504252301list_o @ S )
=> ? [F5: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F5 @ top_top_set_nat )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_614_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_615_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_616_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_617_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_618_inj__image__subset__iff,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ ( image_4575287668734308173list_o @ F @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_619_inj__image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_620_bij__def,axiom,
! [F: nat > option_list_o] :
( ( bij_be1867499782348650780list_o @ F @ top_top_set_nat @ top_to633166595683317524list_o )
= ( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
& ( ( image_4575287668734308173list_o @ F @ top_top_set_nat )
= top_to633166595683317524list_o ) ) ) ).
% bij_def
thf(fact_621_bij__def,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ( inj_on_nat_nat @ F @ top_top_set_nat )
& ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% bij_def
thf(fact_622_bijI,axiom,
! [F: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( ( image_4575287668734308173list_o @ F @ top_top_set_nat )
= top_to633166595683317524list_o )
=> ( bij_be1867499782348650780list_o @ F @ top_top_set_nat @ top_to633166595683317524list_o ) ) ) ).
% bijI
thf(fact_623_bijI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI
thf(fact_624_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_625_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_626_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_627_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_628_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_629_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_630_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_631_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_632_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_633_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_634_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_635_SUP__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D2: nat > nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_636_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_637_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > option_list_o,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( inj_on1630180835328728801list_o @ G @ ( image_nat_nat @ F @ A2 ) )
= ( inj_on1630180835328728801list_o @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_638_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_639_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_640_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_641_Int__greatest,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_642_Int__absorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_643_Int__absorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_644_Int__lower2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_645_Int__lower1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_646_Int__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_647_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_648_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( image_nat_nat @ F @ A2 )
= B ) ) ).
% bij_betw_imp_surj_on
thf(fact_649_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_650_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_651_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_652_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_653_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_654_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_655_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_656_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M10: nat] :
( ( M
= ( suc @ M10 ) )
& ( ord_less_nat @ N2 @ M10 ) ) ) ) ).
% Suc_less_eq2
thf(fact_657_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_658_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_659_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_660_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_661_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_662_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_663_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_664_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_665_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_666_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_667_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X2: nat] : ( R4 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z4: nat] :
( ( R4 @ X2 @ Y3 )
=> ( ( R4 @ Y3 @ Z4 )
=> ( R4 @ X2 @ Z4 ) ) )
=> ( ! [N6: nat] : ( R4 @ N6 @ ( suc @ N6 ) )
=> ( R4 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_668_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N6: nat] :
( ( ord_less_eq_nat @ M @ N6 )
=> ( ( P @ N6 )
=> ( P @ ( suc @ N6 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_669_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N6: nat] :
( ! [M9: nat] :
( ( ord_less_eq_nat @ ( suc @ M9 ) @ N6 )
=> ( P @ M9 ) )
=> ( P @ N6 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_670_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_671_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_672_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_673_Suc__le__D,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
=> ? [M7: nat] :
( M4
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_674_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_675_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_676_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_677_inj__on__Int,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ( inj_on_nat_nat @ F @ A2 )
| ( inj_on_nat_nat @ F @ B ) )
=> ( inj_on_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inj_on_Int
thf(fact_678_inj__on__Int,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_nat] :
( ( ( inj_on1630180835328728801list_o @ F @ A2 )
| ( inj_on1630180835328728801list_o @ F @ B ) )
=> ( inj_on1630180835328728801list_o @ F @ ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inj_on_Int
thf(fact_679_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_680_ex__new__if__finite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A2 )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_681_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_682_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X3: nat,Y2: nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ).
% inj_def
thf(fact_683_inj__def,axiom,
! [F: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
= ( ! [X3: nat,Y2: nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ).
% inj_def
thf(fact_684_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_685_inj__eq,axiom,
! [F: nat > option_list_o,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_686_injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_687_injI,axiom,
! [F: nat > option_list_o] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( inj_on1630180835328728801list_o @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_688_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_689_injD,axiom,
! [F: nat > option_list_o,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_690_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_691_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_692_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y2: nat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: nat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_693_bij__pointE,axiom,
! [F: nat > nat,Y: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: nat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_694_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X2: nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_695_range__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
= S ) ) ).
% range_enumerate
thf(fact_696_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_697_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_698_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > set_nat] :
( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
=> ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ! [N6: nat] :
( ( ( F @ N6 )
= ( F @ ( suc @ N6 ) ) )
=> ( ( F @ ( suc @ N6 ) )
= ( F @ ( suc @ ( suc @ N6 ) ) ) ) )
=> ? [N7: nat] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ N7 )
=> ! [M9: nat] :
( ( ord_less_eq_nat @ M9 @ N7 )
=> ( ( ord_less_nat @ M9 @ N4 )
=> ( ord_less_set_nat @ ( F @ M9 ) @ ( F @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq_nat @ N7 @ N4 )
=> ( ( F @ N7 )
= ( F @ N4 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_699_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ! [N6: nat] :
( ( ( F @ N6 )
= ( F @ ( suc @ N6 ) ) )
=> ( ( F @ ( suc @ N6 ) )
= ( F @ ( suc @ ( suc @ N6 ) ) ) ) )
=> ? [N7: nat] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ N7 )
=> ! [M9: nat] :
( ( ord_less_eq_nat @ M9 @ N7 )
=> ( ( ord_less_nat @ M9 @ N4 )
=> ( ord_less_nat @ ( F @ M9 ) @ ( F @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq_nat @ N7 @ N4 )
=> ( ( F @ N7 )
= ( F @ N4 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_700_SUP__eq,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_701_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_702_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 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_703_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_704_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_705_finite__image__iff,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( finite7007496012504252301list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_706_finite__image__iff,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_707_finite__imageD,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( finite7007496012504252301list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
=> ( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_708_finite__imageD,axiom,
! [F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_709_subset__image__inj,axiom,
! [S: set_option_list_o,F: nat > option_list_o,T2: set_nat] :
( ( ord_le1162937763994921316list_o @ S @ ( image_4575287668734308173list_o @ F @ T2 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T2 )
& ( inj_on1630180835328728801list_o @ F @ U2 )
& ( S
= ( image_4575287668734308173list_o @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_710_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T2 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T2 )
& ( inj_on_nat_nat @ F @ U2 )
& ( S
= ( image_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_711_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ F @ B ) )
= ( A2 = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_712_inj__on__image__eq__iff,axiom,
! [F: nat > option_list_o,C: set_nat,A2: set_nat,B: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( ( image_4575287668734308173list_o @ F @ A2 )
= ( image_4575287668734308173list_o @ F @ B ) )
= ( A2 = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_713_inj__on__image__mem__iff,axiom,
! [F: nat > option_list_o,B: set_nat,A: nat,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ B )
=> ( ( member_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_option_list_o @ ( F @ A ) @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_714_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( member_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_715_bij__betw__subset,axiom,
! [F: nat > nat,A2: set_nat,A8: set_nat,B: set_nat,B11: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ A8 )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ( image_nat_nat @ F @ B )
= B11 )
=> ( bij_betw_nat_nat @ F @ B @ B11 ) ) ) ) ).
% bij_betw_subset
thf(fact_716_bij__betw__byWitness,axiom,
! [A2: set_nat,F6: nat > nat,F: nat > nat,A8: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F6 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A8 )
=> ( ( F @ ( F6 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A8 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F6 @ A8 ) @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ A8 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_717_inj__on__image,axiom,
! [F: nat > nat,A2: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_718_inj__on__image,axiom,
! [F: nat > option_list_o,A2: set_set_nat] :
( ( inj_on1630180835328728801list_o @ F @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on9154615605479958667list_o @ ( image_4575287668734308173list_o @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_719_card__image,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_720_card__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_721_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N2: nat,N8: nat] :
( ! [N6: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N6 ) ) @ ( F @ N6 ) )
=> ( ( ord_less_eq_nat @ N2 @ N8 )
=> ( ord_less_eq_set_nat @ ( F @ N8 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_722_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N8: nat] :
( ! [N6: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N6 ) ) @ ( F @ N6 ) )
=> ( ( ord_less_eq_nat @ N2 @ N8 )
=> ( ord_less_eq_nat @ ( F @ N8 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_723_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N2: nat,N8: nat] :
( ! [N6: nat] : ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N8 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N8 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_724_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N8: nat] :
( ! [N6: nat] : ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N8 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N8 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_725_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N6: nat] : ( ord_less_nat @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_726_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N8: nat] :
( ! [N6: nat] : ( ord_less_nat @ ( F @ N6 ) @ ( F @ ( suc @ N6 ) ) )
=> ( ( ord_less_nat @ N2 @ N8 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N8 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_727_inj__on__imp__bij__betw,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( bij_be1867499782348650780list_o @ F @ A2 @ ( image_4575287668734308173list_o @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_728_inj__on__imp__bij__betw,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ ( image_nat_nat @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_729_bij__betw__imageI,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ( image_4575287668734308173list_o @ F @ A2 )
= B )
=> ( bij_be1867499782348650780list_o @ F @ A2 @ B ) ) ) ).
% bij_betw_imageI
thf(fact_730_bij__betw__imageI,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( image_nat_nat @ F @ A2 )
= B )
=> ( bij_betw_nat_nat @ F @ A2 @ B ) ) ) ).
% bij_betw_imageI
thf(fact_731_bij__betw__def,axiom,
( bij_be1867499782348650780list_o
= ( ^ [F2: nat > option_list_o,A5: set_nat,B3: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A5 )
& ( ( image_4575287668734308173list_o @ F2 @ A5 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_732_bij__betw__def,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
& ( ( image_nat_nat @ F2 @ A5 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_733_image__strict__mono,axiom,
! [F: nat > nat,B: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% image_strict_mono
thf(fact_734_image__strict__mono,axiom,
! [F: nat > option_list_o,B: set_nat,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ B )
=> ( ( ord_less_set_nat @ A2 @ B )
=> ( ord_le4476516537835661936list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ ( image_4575287668734308173list_o @ F @ B ) ) ) ) ).
% image_strict_mono
thf(fact_735_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_736_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_737_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N6: nat] :
( ( ord_less_eq_nat @ I @ N6 )
=> ( ( ord_less_nat @ N6 @ J2 )
=> ( ( P @ N6 )
=> ( P @ ( suc @ N6 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_738_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N6: nat] :
( ( ord_less_eq_nat @ I @ N6 )
=> ( ( ord_less_nat @ N6 @ J2 )
=> ( ( P @ ( suc @ N6 ) )
=> ( P @ N6 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_739_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_740_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_741_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_742_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_743_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_744_monoD,axiom,
! [F: set_nat > set_nat,X: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_745_monoD,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_746_monoD,axiom,
! [F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_747_monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_748_monoE,axiom,
! [F: set_nat > set_nat,X: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_749_monoE,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_750_monoE,axiom,
! [F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_751_monoE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_752_monoI,axiom,
! [F: set_nat > set_nat] :
( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_753_monoI,axiom,
! [F: set_nat > nat] :
( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_754_monoI,axiom,
! [F: nat > set_nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_755_monoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_756_mono__imp__mono__on,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( monoto1748750089227133045et_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_757_mono__imp__mono__on,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( monoto2923694778811248831at_nat @ A2 @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_758_mono__imp__mono__on,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( monoto6489329683466618047et_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_759_mono__imp__mono__on,axiom,
! [F: nat > nat,A2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_760_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_761_linorder__injI,axiom,
! [F: nat > option_list_o] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) )
=> ( inj_on1630180835328728801list_o @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_762_strict__monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_763_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_764_strict__mono__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_765_strict__mono__less,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_766_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A2 = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_767_bij__is__inj,axiom,
! [F: nat > option_list_o] :
( ( bij_be1867499782348650780list_o @ F @ top_top_set_nat @ top_to633166595683317524list_o )
=> ( inj_on1630180835328728801list_o @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_768_bij__is__inj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_769_inj__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( inj_on_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat ) ) ).
% inj_enumerate
thf(fact_770_finite__surj__inj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F @ A2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_771_inj__on__finite,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ B )
=> ( ( finite7007496012504252301list_o @ B )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_772_inj__on__finite,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_773_endo__inj__surj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A2 )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( image_nat_nat @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_774_card__image__le,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_775_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F: nat > option_list_o] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on1630180835328728801list_o @ F @ A2 )
= ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_776_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_777_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F: nat > option_list_o] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on1630180835328728801list_o @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_778_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_779_pigeonhole,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( ord_less_nat @ ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on1630180835328728801list_o @ F @ A2 ) ) ).
% pigeonhole
thf(fact_780_pigeonhole,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_781_bij__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).
% bij_enumerate
thf(fact_782_strict__mono__less__eq,axiom,
! [F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_nat @ ord_less_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_783_strict__mono__less__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_784_mono__strict__invE,axiom,
! [F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_785_mono__strict__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_786_strict__mono__mono,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_set_nat @ F )
=> ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% strict_mono_mono
thf(fact_787_strict__mono__mono,axiom,
! [F: set_nat > nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ F )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_788_strict__mono__mono,axiom,
! [F: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_nat @ ord_less_set_nat @ F )
=> ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% strict_mono_mono
thf(fact_789_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_790_mono__invE,axiom,
! [F: nat > set_nat,X: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_791_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_792_enumerate__step,axiom,
! [S: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N2 ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N2 ) ) ) ) ).
% enumerate_step
thf(fact_793_strict__mono__imp__inj__on,axiom,
! [F: nat > nat,A2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% strict_mono_imp_inj_on
thf(fact_794_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_795_infinite__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [R: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R )
& ! [N4: nat] : ( member_nat @ ( R @ N4 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_796_surj__card__le,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 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_797_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_option_list_o,F: nat > option_list_o] :
( ( finite_finite_nat @ S )
=> ( ( finite7007496012504252301list_o @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite3362998479529755404list_o @ T2 ) )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ S ) @ T2 )
=> ( ( ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ T2 )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ S )
& ( ( F @ Y2 )
= X3 ) ) ) )
= ( inj_on1630180835328728801list_o @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_798_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S ) @ T2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ S )
& ( ( F @ Y2 )
= X3 ) ) ) )
= ( inj_on_nat_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_799_card__bij__eq,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o,G: option_list_o > nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ B )
=> ( ( inj_on2456431687840576515_o_nat @ G @ B )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ G @ B ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite7007496012504252301list_o @ B )
=> ( ( finite_card_nat @ A2 )
= ( finite3362998479529755404list_o @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_800_card__bij__eq,axiom,
! [F: option_list_o > nat,A2: set_option_list_o,B: set_nat,G: nat > option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ F @ A2 ) @ B )
=> ( ( inj_on1630180835328728801list_o @ G @ B )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ G @ B ) @ A2 )
=> ( ( finite7007496012504252301list_o @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( finite3362998479529755404list_o @ A2 )
= ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_801_card__bij__eq,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ( inj_on_nat_nat @ G @ B )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_802_strict__mono__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
= ( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_803_incseq__Suc__iff,axiom,
! [F: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
= ( ! [N5: nat] : ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_804_incseq__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
= ( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_805_incseq__SucI,axiom,
! [X4: nat > set_nat] :
( ! [N6: nat] : ( ord_less_eq_set_nat @ ( X4 @ N6 ) @ ( X4 @ ( suc @ N6 ) ) )
=> ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ X4 ) ) ).
% incseq_SucI
thf(fact_806_incseq__SucI,axiom,
! [X4: nat > nat] :
( ! [N6: nat] : ( ord_less_eq_nat @ ( X4 @ N6 ) @ ( X4 @ ( suc @ N6 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X4 ) ) ).
% incseq_SucI
thf(fact_807_incseq__SucD,axiom,
! [A2: nat > set_nat,I: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ A2 )
=> ( ord_less_eq_set_nat @ ( A2 @ I ) @ ( A2 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_808_incseq__SucD,axiom,
! [A2: nat > nat,I: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ A2 )
=> ( ord_less_eq_nat @ ( A2 @ I ) @ ( A2 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_809_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_810_inf_Obounded__iff,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_set_nat @ A @ B4 )
& ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_811_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_812_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_813_le__inf__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_814_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_815_mono__Int,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_Int
thf(fact_816_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_817_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_818_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_819_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_820_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_821_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_822_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_823_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_824_le__infE,axiom,
! [X: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_825_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_826_le__infI,axiom,
! [X: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B4 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_827_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_828_inf__mono,axiom,
! [A: set_nat,C2: set_nat,B4: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ C2 @ D3 ) ) ) ) ).
% inf_mono
thf(fact_829_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D3 ) ) ) ) ).
% inf_mono
thf(fact_830_le__infI1,axiom,
! [A: set_nat,X: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_831_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_832_le__infI2,axiom,
! [B4: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_833_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_834_inf_OorderE,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( A
= ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_835_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_836_inf_OorderI,axiom,
! [A: set_nat,B4: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_837_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_838_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_839_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,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z4 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_840_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_841_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_842_inf_Oabsorb1,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( inf_inf_set_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_843_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_844_inf_Oabsorb2,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( inf_inf_set_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_845_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_846_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_847_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_848_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_849_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_850_inf_OboundedE,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_851_inf_OboundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_852_inf_OboundedI,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_853_inf_OboundedI,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_854_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z3 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_855_inf__greatest,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_856_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B10: set_nat] :
( A7
= ( inf_inf_set_nat @ A7 @ B10 ) ) ) ) ).
% inf.order_iff
thf(fact_857_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B10: nat] :
( A7
= ( inf_inf_nat @ A7 @ B10 ) ) ) ) ).
% inf.order_iff
thf(fact_858_inf_Ocobounded1,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_859_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_860_inf_Ocobounded2,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_861_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_862_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B10: set_nat] :
( ( inf_inf_set_nat @ A7 @ B10 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_863_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B10: nat] :
( ( inf_inf_nat @ A7 @ B10 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_864_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B10: set_nat,A7: set_nat] :
( ( inf_inf_set_nat @ A7 @ B10 )
= B10 ) ) ) ).
% inf.absorb_iff2
thf(fact_865_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B10: nat,A7: nat] :
( ( inf_inf_nat @ A7 @ B10 )
= B10 ) ) ) ).
% inf.absorb_iff2
thf(fact_866_inf_OcoboundedI1,axiom,
! [A: set_nat,C2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_867_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_868_inf_OcoboundedI2,axiom,
! [B4: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_869_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_870_inf_Ostrict__coboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_871_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_872_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A7: nat,B10: nat] :
( ( A7
= ( inf_inf_nat @ A7 @ B10 ) )
& ( A7 != B10 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_873_inf_Ostrict__boundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_874_inf_Oabsorb4,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_875_inf_Oabsorb3,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_876_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_877_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_878_strict__mono__leD,axiom,
! [R2: set_nat > set_nat,M: set_nat,N2: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_set_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ M @ N2 )
=> ( ord_less_eq_set_nat @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_879_strict__mono__leD,axiom,
! [R2: set_nat > nat,M: set_nat,N2: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_880_strict__mono__leD,axiom,
! [R2: nat > set_nat,M: nat,N2: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_nat @ ord_less_set_nat @ R2 )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_set_nat @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_881_strict__mono__leD,axiom,
! [R2: nat > nat,M: nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_882_incseqD,axiom,
! [F: nat > set_nat,I: nat,J2: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% incseqD
thf(fact_883_incseqD,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% incseqD
thf(fact_884_incseq__def,axiom,
! [X4: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ X4 )
= ( ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_set_nat @ ( X4 @ M5 ) @ ( X4 @ N5 ) ) ) ) ) ).
% incseq_def
thf(fact_885_incseq__def,axiom,
! [X4: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X4 )
= ( ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_nat @ ( X4 @ M5 ) @ ( X4 @ N5 ) ) ) ) ) ).
% incseq_def
thf(fact_886_Schroeder__Bernstein,axiom,
! [F: nat > option_list_o,A2: set_nat,B: set_option_list_o,G: option_list_o > nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A2 ) @ B )
=> ( ( inj_on2456431687840576515_o_nat @ G @ B )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ G @ B ) @ A2 )
=> ? [H2: nat > option_list_o] : ( bij_be1867499782348650780list_o @ H2 @ A2 @ B ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_887_Schroeder__Bernstein,axiom,
! [F: option_list_o > nat,A2: set_option_list_o,B: set_nat,G: nat > option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ F @ A2 ) @ B )
=> ( ( inj_on1630180835328728801list_o @ G @ B )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ G @ B ) @ A2 )
=> ? [H2: option_list_o > nat] : ( bij_be2693750634860498494_o_nat @ H2 @ A2 @ B ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_888_Schroeder__Bernstein,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ( inj_on_nat_nat @ G @ B )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B ) @ A2 )
=> ? [H2: nat > nat] : ( bij_betw_nat_nat @ H2 @ A2 @ B ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_889_ex__subset__image__inj,axiom,
! [F: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ? [T5: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T5 @ ( image_4575287668734308173list_o @ F @ S ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on1630180835328728801list_o @ F @ T5 )
& ( P @ ( image_4575287668734308173list_o @ F @ T5 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_890_ex__subset__image__inj,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on_nat_nat @ F @ T5 )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_891_all__subset__image__inj,axiom,
! [F: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ! [T5: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T5 @ ( image_4575287668734308173list_o @ F @ S ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on1630180835328728801list_o @ F @ T5 ) )
=> ( P @ ( image_4575287668734308173list_o @ F @ T5 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_892_all__subset__image__inj,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on_nat_nat @ F @ T5 ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_893_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X9: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X9 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X9 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_894_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X9: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X9 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X9 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_895_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_896_bijI_H,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: nat] :
? [X6: nat] :
( Y3
= ( F @ X6 ) )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_897_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_898_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_899_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_900_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_901_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_902_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_903_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_904_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_905_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_906_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_907_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_908_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_909_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
=> ( ~ ( P @ N6 )
=> ? [M9: nat] :
( ( ord_less_nat @ M9 @ N6 )
& ~ ( P @ M9 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_910_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_911_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_912_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_913_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_914_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_915_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_916_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_917_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_918_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_919_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_920_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_921_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N6: nat] :
( ( P @ N6 )
=> ( P @ ( suc @ N6 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_922_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_923_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N6: nat] :
( ( P @ ( suc @ N6 ) )
=> ( P @ N6 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_924_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_925_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_926_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_927_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M7: nat] :
( N2
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_928_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_929_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M7: nat] :
( N2
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_930_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_931_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_932_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_933_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_934_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_935_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_936_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_937_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_938_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_939_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_940_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_941_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_942_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_943_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_944_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_945_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_946_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_947_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_948_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_949_to__nat__on__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( counta4844910239362777137on_nat @ S ) @ S @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) ) ) ).
% to_nat_on_finite
thf(fact_950_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_951_incseq__imp__monoseq,axiom,
! [X4: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ X4 )
=> ( topolo7278393974255667507et_nat @ X4 ) ) ).
% incseq_imp_monoseq
thf(fact_952_incseq__imp__monoseq,axiom,
! [X4: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X4 )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% incseq_imp_monoseq
thf(fact_953_inj__on__image__Fpow,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_954_inj__on__image__Fpow,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( inj_on9154615605479958667list_o @ ( image_4575287668734308173list_o @ F ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_955_bij__betw__from__nat__into__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( counta7321652538601044515to_nat @ S ) @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) @ S ) ) ).
% bij_betw_from_nat_into_finite
thf(fact_956_mono__SucI1,axiom,
! [X4: nat > set_nat] :
( ! [N6: nat] : ( ord_less_eq_set_nat @ ( X4 @ N6 ) @ ( X4 @ ( suc @ N6 ) ) )
=> ( topolo7278393974255667507et_nat @ X4 ) ) ).
% mono_SucI1
thf(fact_957_mono__SucI1,axiom,
! [X4: nat > nat] :
( ! [N6: nat] : ( ord_less_eq_nat @ ( X4 @ N6 ) @ ( X4 @ ( suc @ N6 ) ) )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% mono_SucI1
thf(fact_958_mono__SucI2,axiom,
! [X4: nat > set_nat] :
( ! [N6: nat] : ( ord_less_eq_set_nat @ ( X4 @ ( suc @ N6 ) ) @ ( X4 @ N6 ) )
=> ( topolo7278393974255667507et_nat @ X4 ) ) ).
% mono_SucI2
thf(fact_959_mono__SucI2,axiom,
! [X4: nat > nat] :
( ! [N6: nat] : ( ord_less_eq_nat @ ( X4 @ ( suc @ N6 ) ) @ ( X4 @ N6 ) )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% mono_SucI2
thf(fact_960_monoseq__Suc,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X9: nat > set_nat] :
( ! [N5: nat] : ( ord_less_eq_set_nat @ ( X9 @ N5 ) @ ( X9 @ ( suc @ N5 ) ) )
| ! [N5: nat] : ( ord_less_eq_set_nat @ ( X9 @ ( suc @ N5 ) ) @ ( X9 @ N5 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_961_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X9: nat > nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( X9 @ N5 ) @ ( X9 @ ( suc @ N5 ) ) )
| ! [N5: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N5 ) ) @ ( X9 @ N5 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_962_monoseq__def,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X9: nat > set_nat] :
( ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_set_nat @ ( X9 @ M5 ) @ ( X9 @ N5 ) ) )
| ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_set_nat @ ( X9 @ N5 ) @ ( X9 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_963_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X9: nat > nat] :
( ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_nat @ ( X9 @ M5 ) @ ( X9 @ N5 ) ) )
| ! [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
=> ( ord_less_eq_nat @ ( X9 @ N5 ) @ ( X9 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_964_monoI2,axiom,
! [X4: nat > set_nat] :
( ! [M7: nat,N6: nat] :
( ( ord_less_eq_nat @ M7 @ N6 )
=> ( ord_less_eq_set_nat @ ( X4 @ N6 ) @ ( X4 @ M7 ) ) )
=> ( topolo7278393974255667507et_nat @ X4 ) ) ).
% monoI2
thf(fact_965_monoI2,axiom,
! [X4: nat > nat] :
( ! [M7: nat,N6: nat] :
( ( ord_less_eq_nat @ M7 @ N6 )
=> ( ord_less_eq_nat @ ( X4 @ N6 ) @ ( X4 @ M7 ) ) )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% monoI2
thf(fact_966_monoI1,axiom,
! [X4: nat > set_nat] :
( ! [M7: nat,N6: nat] :
( ( ord_less_eq_nat @ M7 @ N6 )
=> ( ord_less_eq_set_nat @ ( X4 @ M7 ) @ ( X4 @ N6 ) ) )
=> ( topolo7278393974255667507et_nat @ X4 ) ) ).
% monoI1
thf(fact_967_monoI1,axiom,
! [X4: nat > nat] :
( ! [M7: nat,N6: nat] :
( ( ord_less_eq_nat @ M7 @ N6 )
=> ( ord_less_eq_nat @ ( X4 @ M7 ) @ ( X4 @ N6 ) ) )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% monoI1
thf(fact_968_Fpow__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A2 ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_969_image__Fpow__mono,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_970_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_971_countable__enum__cases,axiom,
! [S: set_nat] :
( ( counta1168086296615599829le_nat @ S )
=> ( ( ( finite_finite_nat @ S )
=> ! [F5: nat > nat] :
~ ( bij_betw_nat_nat @ F5 @ S @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) ) )
=> ~ ( ~ ( finite_finite_nat @ S )
=> ! [F5: nat > nat] :
~ ( bij_betw_nat_nat @ F5 @ S @ top_top_set_nat ) ) ) ) ).
% countable_enum_cases
thf(fact_972_lfp__ordinal__induct__set,axiom,
! [F: set_nat > set_nat,P: set_nat > $o] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ! [S6: set_nat] :
( ( P @ S6 )
=> ( P @ ( F @ S6 ) ) )
=> ( ! [M6: set_set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ M6 )
=> ( P @ X6 ) )
=> ( P @ ( comple7399068483239264473et_nat @ M6 ) ) )
=> ( P @ ( comple7975543026063415949et_nat @ F ) ) ) ) ) ).
% lfp_ordinal_induct_set
thf(fact_973_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_974_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_975_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_976_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_977_countable__iff__bij,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( counta1168086296615599829le_nat @ A2 )
= ( counta1168086296615599829le_nat @ B ) ) ) ).
% countable_iff_bij
thf(fact_978_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_979_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_980_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_981_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_982_from__nat__into__inj__infinite,axiom,
! [A2: set_nat,M: nat,N2: nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( ( counta7321652538601044515to_nat @ A2 @ M )
= ( counta7321652538601044515to_nat @ A2 @ N2 ) )
= ( M = N2 ) ) ) ) ).
% from_nat_into_inj_infinite
thf(fact_983_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_984_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_985_to__nat__on__from__nat__into__infinite,axiom,
! [A2: set_nat,N2: nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( counta4844910239362777137on_nat @ A2 @ ( counta7321652538601044515to_nat @ A2 @ N2 ) )
= N2 ) ) ) ).
% to_nat_on_from_nat_into_infinite
thf(fact_986_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_987_uncountable__infinite,axiom,
! [A2: set_nat] :
( ~ ( counta1168086296615599829le_nat @ A2 )
=> ~ ( finite_finite_nat @ A2 ) ) ).
% uncountable_infinite
thf(fact_988_countable__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( counta1168086296615599829le_nat @ S ) ) ).
% countable_finite
thf(fact_989_countable__Collect__finite,axiom,
counta3299167949292459659et_nat @ ( collect_set_nat @ finite_finite_nat ) ).
% countable_Collect_finite
thf(fact_990_countableI_H,axiom,
! [F: nat > nat,S: set_nat] :
( ( inj_on_nat_nat @ F @ S )
=> ( counta1168086296615599829le_nat @ S ) ) ).
% countableI'
thf(fact_991_countableI_H,axiom,
! [F: nat > option_list_o,S: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ S )
=> ( counta1168086296615599829le_nat @ S ) ) ).
% countableI'
thf(fact_992_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_993_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_994_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_995_mult__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_996_mult__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_997_mult__le__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_998_lfp__greatest,axiom,
! [F: set_nat > set_nat,A2: set_nat] :
( ! [U3: set_nat] :
( ( ord_less_eq_set_nat @ ( F @ U3 ) @ U3 )
=> ( ord_less_eq_set_nat @ A2 @ U3 ) )
=> ( ord_less_eq_set_nat @ A2 @ ( comple7975543026063415949et_nat @ F ) ) ) ).
% lfp_greatest
thf(fact_999_lfp__lowerbound,axiom,
! [F: set_nat > set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( F @ A2 ) @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ A2 ) ) ).
% lfp_lowerbound
thf(fact_1000_lfp__mono,axiom,
! [F: set_nat > set_nat,G: set_nat > set_nat] :
( ! [Z6: set_nat] : ( ord_less_eq_set_nat @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ ( comple7975543026063415949et_nat @ G ) ) ) ).
% lfp_mono
thf(fact_1001_countable__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( counta1168086296615599829le_nat @ B )
=> ( counta1168086296615599829le_nat @ A2 ) ) ) ).
% countable_subset
thf(fact_1002_uncountable__bij__betw,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ~ ( counta1168086296615599829le_nat @ B )
=> ~ ( counta1168086296615599829le_nat @ A2 ) ) ) ).
% uncountable_bij_betw
thf(fact_1003_countableI__bij2,axiom,
! [F: nat > nat,B: set_nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ B @ A2 )
=> ( ( counta1168086296615599829le_nat @ A2 )
=> ( counta1168086296615599829le_nat @ B ) ) ) ).
% countableI_bij2
thf(fact_1004_countableI__bij1,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B )
=> ( ( counta1168086296615599829le_nat @ A2 )
=> ( counta1168086296615599829le_nat @ B ) ) ) ).
% countableI_bij1
thf(fact_1005_countableI__bij,axiom,
! [F: nat > nat,C: set_nat,S: set_nat] :
( ( bij_betw_nat_nat @ F @ C @ S )
=> ( counta1168086296615599829le_nat @ S ) ) ).
% countableI_bij
thf(fact_1006_countableE__bij,axiom,
! [S: set_nat] :
( ( counta1168086296615599829le_nat @ S )
=> ~ ! [F5: nat > nat,C4: set_nat] :
~ ( bij_betw_nat_nat @ F5 @ C4 @ S ) ) ).
% countableE_bij
thf(fact_1007_countable__image__eq,axiom,
! [F: nat > nat,S: set_nat] :
( ( counta1168086296615599829le_nat @ ( image_nat_nat @ F @ S ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( ( image_nat_nat @ F @ S )
= ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% countable_image_eq
thf(fact_1008_countable__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ( counta1168086296615599829le_nat @ B )
& ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) ) )
= ( ? [A9: set_nat] :
( ( counta1168086296615599829le_nat @ A9 )
& ( ord_less_eq_set_nat @ A9 @ A2 )
& ( B
= ( image_nat_nat @ F @ A9 ) ) ) ) ) ).
% countable_subset_image
thf(fact_1009_ex__countable__subset__image,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_countable_subset_image
thf(fact_1010_all__countable__subset__image,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_countable_subset_image
thf(fact_1011_infinite__countable__subset_H,axiom,
! [X4: set_nat] :
( ~ ( finite_finite_nat @ X4 )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ X4 )
& ( counta1168086296615599829le_nat @ C4 )
& ~ ( finite_finite_nat @ C4 ) ) ) ).
% infinite_countable_subset'
thf(fact_1012_countable__image__inj__on,axiom,
! [F: nat > nat,A2: set_nat] :
( ( counta1168086296615599829le_nat @ ( image_nat_nat @ F @ A2 ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( counta1168086296615599829le_nat @ A2 ) ) ) ).
% countable_image_inj_on
thf(fact_1013_countable__image__inj__on,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( counta4250636581802663583list_o @ ( image_4575287668734308173list_o @ F @ A2 ) )
=> ( ( inj_on1630180835328728801list_o @ F @ A2 )
=> ( counta1168086296615599829le_nat @ A2 ) ) ) ).
% countable_image_inj_on
thf(fact_1014_countable__image__inj__eq,axiom,
! [F: nat > nat,S: set_nat] :
( ( inj_on_nat_nat @ F @ S )
=> ( ( counta1168086296615599829le_nat @ ( image_nat_nat @ F @ S ) )
= ( counta1168086296615599829le_nat @ S ) ) ) ).
% countable_image_inj_eq
thf(fact_1015_countable__image__inj__eq,axiom,
! [F: nat > option_list_o,S: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ S )
=> ( ( counta4250636581802663583list_o @ ( image_4575287668734308173list_o @ F @ S ) )
= ( counta1168086296615599829le_nat @ S ) ) ) ).
% countable_image_inj_eq
thf(fact_1016_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1017_mult__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1018_mult__less__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1019_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1020_to__nat__on__surj,axiom,
! [A2: set_nat,N2: nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( counta4844910239362777137on_nat @ A2 @ X2 )
= N2 ) ) ) ) ).
% to_nat_on_surj
thf(fact_1021_all__countable__subset__image__inj,axiom,
! [F: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ! [T5: set_option_list_o] :
( ( ( counta4250636581802663583list_o @ T5 )
& ( ord_le1162937763994921316list_o @ T5 @ ( image_4575287668734308173list_o @ F @ S ) ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on1630180835328728801list_o @ F @ T5 ) )
=> ( P @ ( image_4575287668734308173list_o @ F @ T5 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_1022_all__countable__subset__image__inj,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) ) )
=> ( P @ T5 ) ) )
= ( ! [T5: set_nat] :
( ( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on_nat_nat @ F @ T5 ) )
=> ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_1023_ex__countable__subset__image__inj,axiom,
! [F: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ? [T5: set_option_list_o] :
( ( counta4250636581802663583list_o @ T5 )
& ( ord_le1162937763994921316list_o @ T5 @ ( image_4575287668734308173list_o @ F @ S ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on1630180835328728801list_o @ F @ T5 )
& ( P @ ( image_4575287668734308173list_o @ F @ T5 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_1024_ex__countable__subset__image__inj,axiom,
! [F: nat > nat,S: set_nat,P: set_nat > $o] :
( ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ ( image_nat_nat @ F @ S ) )
& ( P @ T5 ) ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( inj_on_nat_nat @ F @ T5 )
& ( P @ ( image_nat_nat @ F @ T5 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_1025_countable__image__eq__inj,axiom,
! [F: nat > nat,S: set_nat] :
( ( counta1168086296615599829le_nat @ ( image_nat_nat @ F @ S ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( ( image_nat_nat @ F @ S )
= ( image_nat_nat @ F @ T5 ) )
& ( inj_on_nat_nat @ F @ T5 ) ) ) ) ).
% countable_image_eq_inj
thf(fact_1026_countable__image__eq__inj,axiom,
! [F: nat > option_list_o,S: set_nat] :
( ( counta4250636581802663583list_o @ ( image_4575287668734308173list_o @ F @ S ) )
= ( ? [T5: set_nat] :
( ( counta1168086296615599829le_nat @ T5 )
& ( ord_less_eq_set_nat @ T5 @ S )
& ( ( image_4575287668734308173list_o @ F @ S )
= ( image_4575287668734308173list_o @ F @ T5 ) )
& ( inj_on1630180835328728801list_o @ F @ T5 ) ) ) ) ).
% countable_image_eq_inj
thf(fact_1027_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1028_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1029_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1030_countable__infiniteE_H,axiom,
! [A2: set_nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ~ ! [G4: nat > nat] :
~ ( bij_betw_nat_nat @ G4 @ top_top_set_nat @ A2 ) ) ) ).
% countable_infiniteE'
thf(fact_1031_countableE__infinite,axiom,
! [S: set_nat] :
( ( counta1168086296615599829le_nat @ S )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ! [E3: nat > nat] :
~ ( bij_betw_nat_nat @ E3 @ S @ top_top_set_nat ) ) ) ).
% countableE_infinite
thf(fact_1032_def__lfp__unfold,axiom,
! [H: set_nat,F: set_nat > set_nat] :
( ( H
= ( comple7975543026063415949et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( H
= ( F @ H ) ) ) ) ).
% def_lfp_unfold
thf(fact_1033_lfp__fixpoint,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( F @ ( comple7975543026063415949et_nat @ F ) )
= ( comple7975543026063415949et_nat @ F ) ) ) ).
% lfp_fixpoint
thf(fact_1034_lfp__unfold,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( comple7975543026063415949et_nat @ F )
= ( F @ ( comple7975543026063415949et_nat @ F ) ) ) ) ).
% lfp_unfold
thf(fact_1035_lfp__eqI,axiom,
! [F4: set_nat > set_nat,X: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z4: set_nat] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq_set_nat @ X @ Z4 ) )
=> ( ( comple7975543026063415949et_nat @ F4 )
= X ) ) ) ) ).
% lfp_eqI
thf(fact_1036_subset__range__from__nat__into,axiom,
! [A2: set_nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ ( counta7321652538601044515to_nat @ A2 ) @ top_top_set_nat ) ) ) ).
% subset_range_from_nat_into
thf(fact_1037_countable__as__injective__image,axiom,
! [A2: set_option_list_o] :
( ( counta4250636581802663583list_o @ A2 )
=> ( ~ ( finite7007496012504252301list_o @ A2 )
=> ~ ! [F5: nat > option_list_o] :
( ( A2
= ( image_4575287668734308173list_o @ F5 @ top_top_set_nat ) )
=> ~ ( inj_on1630180835328728801list_o @ F5 @ top_top_set_nat ) ) ) ) ).
% countable_as_injective_image
thf(fact_1038_countable__as__injective__image,axiom,
! [A2: set_nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ~ ! [F5: nat > nat] :
( ( A2
= ( image_nat_nat @ F5 @ top_top_set_nat ) )
=> ~ ( inj_on_nat_nat @ F5 @ top_top_set_nat ) ) ) ) ).
% countable_as_injective_image
thf(fact_1039_image__to__nat__on,axiom,
! [A2: set_nat] :
( ( counta1168086296615599829le_nat @ A2 )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( image_nat_nat @ ( counta4844910239362777137on_nat @ A2 ) @ A2 )
= top_top_set_nat ) ) ) ).
% image_to_nat_on
thf(fact_1040_bij__betw__from__nat__into,axiom,
! [S: set_nat] :
( ( counta1168086296615599829le_nat @ S )
=> ( ~ ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( counta7321652538601044515to_nat @ S ) @ top_top_set_nat @ S ) ) ) ).
% bij_betw_from_nat_into
thf(fact_1041_to__nat__on__infinite,axiom,
! [S: set_nat] :
( ( counta1168086296615599829le_nat @ S )
=> ( ~ ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( counta4844910239362777137on_nat @ S ) @ S @ top_top_set_nat ) ) ) ).
% to_nat_on_infinite
thf(fact_1042_def__lfp__induct,axiom,
! [A2: set_nat,F: set_nat > set_nat,P: set_nat] :
( ( A2
= ( comple7975543026063415949et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ ( F @ ( inf_inf_set_nat @ A2 @ P ) ) @ P )
=> ( ord_less_eq_set_nat @ A2 @ P ) ) ) ) ).
% def_lfp_induct
thf(fact_1043_lfp__induct,axiom,
! [F: set_nat > set_nat,P: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ ( F @ ( inf_inf_set_nat @ ( comple7975543026063415949et_nat @ F ) @ P ) ) @ P )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ P ) ) ) ).
% lfp_induct
thf(fact_1044_lfp__ordinal__induct,axiom,
! [F: set_nat > set_nat,P: set_nat > $o] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ! [S6: set_nat] :
( ( P @ S6 )
=> ( ( ord_less_eq_set_nat @ S6 @ ( comple7975543026063415949et_nat @ F ) )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M6: set_set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ M6 )
=> ( P @ X6 ) )
=> ( P @ ( comple7399068483239264473et_nat @ M6 ) ) )
=> ( P @ ( comple7975543026063415949et_nat @ F ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_1045_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1046_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1047_mult__mono,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1048_mult__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1049_mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_1050_mult__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_1051_split__mult__neg__le,axiom,
! [A: nat,B4: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B4 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1052_mult__nonneg__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1053_mult__nonneg__nonpos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1054_mult__nonpos__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1055_mult__nonneg__nonpos2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1056_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1057_mult__neg__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1058_mult__pos__neg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1059_mult__pos__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_pos_pos
thf(fact_1060_mult__pos__neg2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1061_zero__less__mult__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_1062_zero__less__mult__pos2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B4 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1063_mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1064_mult__strict__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1065_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1066_inj__on__mult,axiom,
! [A: nat,A2: set_nat] :
( ( A != zero_zero_nat )
=> ( inj_on_nat_nat @ ( times_times_nat @ A ) @ A2 ) ) ).
% inj_on_mult
thf(fact_1067_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1068_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1069_mult__less__le__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1070_mult__le__less__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1071_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1072_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1073_mult__strict__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1074_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1075_mult__strict__mono,axiom,
! [A: nat,B4: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1076_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1077_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1078_mono__mult,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ A ) ) ) ).
% mono_mult
thf(fact_1079_ccSup__inter__less__eq,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( counta3299167949292459659et_nat @ A2 )
=> ( ( counta3299167949292459659et_nat @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_1080_ccSup__subset__mono,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( counta3299167949292459659et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ) ).
% ccSup_subset_mono
thf(fact_1081_ccSup__mono,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( counta3299167949292459659et_nat @ B )
=> ( ( counta3299167949292459659et_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B )
& ( ord_less_eq_set_nat @ A4 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ) ) ).
% ccSup_mono
thf(fact_1082_ccSup__least,axiom,
! [A2: set_set_nat,Z3: set_nat] :
( ( counta3299167949292459659et_nat @ A2 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z3 ) ) ) ).
% ccSup_least
thf(fact_1083_ccSup__upper,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( counta3299167949292459659et_nat @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% ccSup_upper
thf(fact_1084_ccSup__le__iff,axiom,
! [A2: set_set_nat,B4: set_nat] :
( ( counta3299167949292459659et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B4 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B4 ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_1085_ccSup__upper2,axiom,
! [A2: set_set_nat,U: set_nat,V: set_nat] :
( ( counta3299167949292459659et_nat @ A2 )
=> ( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ) ).
% ccSup_upper2
thf(fact_1086_card__partition,axiom,
! [C: set_set_nat,K: nat] :
( ( finite1152437895449049373et_nat @ C )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C ) )
=> ( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ C )
=> ( ( finite_card_nat @ C3 )
= K ) )
=> ( ! [C1: set_nat,C22: set_nat] :
( ( member_set_nat @ C1 @ C )
=> ( ( member_set_nat @ C22 @ C )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_nat @ C1 @ C22 )
= bot_bot_set_nat ) ) ) )
=> ( ( times_times_nat @ K @ ( finite_card_set_nat @ C ) )
= ( finite_card_nat @ ( comple7399068483239264473et_nat @ C ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1087_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_1088_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_1089_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1090_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1091_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1092_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1093_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1094_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1095_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1096_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1097_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_1098_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1099_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1100_inj__on__empty,axiom,
! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_1101_inj__on__empty,axiom,
! [F: nat > option_list_o] : ( inj_on1630180835328728801list_o @ F @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_1102_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1103_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_1104_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_1105_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1106_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1107_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( ( inf_inf_set_nat @ X3 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1108_empty__in__Fpow,axiom,
! [A2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A2 ) ) ).
% empty_in_Fpow
thf(fact_1109_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1110_monotone__on__empty,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] : ( monotone_on_nat_nat @ bot_bot_set_nat @ Orda @ Ordb @ F ) ).
% monotone_on_empty
thf(fact_1111_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1112_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_1113_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1114_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1115_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1116_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1117_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1118_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_1119_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_1120_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1121_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1122_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1123_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1124_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( X3 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1125_finite__transitivity__chain,axiom,
! [A2: set_nat,R4: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X2: nat] :
~ ( R4 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z4: nat] :
( ( R4 @ X2 @ Y3 )
=> ( ( R4 @ Y3 @ Z4 )
=> ( R4 @ X2 @ Z4 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( R4 @ X2 @ Y6 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1126_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1127_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1128_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1129_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1130_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1131_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1132_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1133_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1134_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1135_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1136_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1137_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1138_bij__betw__empty2,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_1139_bij__betw__empty1,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ bot_bot_set_nat @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_1140_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1141_Union__disjoint,axiom,
! [C: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C ) @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( ( inf_inf_set_nat @ X3 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_1142_less__cSupE,axiom,
! [Y: nat,X4: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
=> ( ( X4 != bot_bot_set_nat )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ) ) ).
% less_cSupE
thf(fact_1143_less__cSupD,axiom,
! [X4: set_nat,Z3: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z3 @ ( complete_Sup_Sup_nat @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( ord_less_nat @ Z3 @ X2 ) ) ) ) ).
% less_cSupD
thf(fact_1144_less__eq__Sup,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1145_cSup__least,axiom,
! [X4: set_set_nat,Z3: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1146_cSup__least,axiom,
! [X4: set_nat,Z3: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1147_cSup__eq__non__empty,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ A ) )
=> ( ! [Y3: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X4 )
=> ( ord_less_eq_set_nat @ X6 @ Y3 ) )
=> ( ord_less_eq_set_nat @ A @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1148_cSup__eq__non__empty,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y3: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X4 )
=> ( ord_less_eq_nat @ X6 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1149_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1150_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1151_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1152_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1153_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1154_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1155_SUP__eq__iff,axiom,
! [I5: set_nat,C2: set_nat,F: nat > set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_set_nat @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= C2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1156_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,M8: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M8 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ M8 ) ) ) ).
% cSUP_least
thf(fact_1157_cSUP__least,axiom,
! [A2: set_nat,F: nat > nat,M8: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M8 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A2 ) ) @ M8 ) ) ) ).
% cSUP_least
thf(fact_1158_finite__Sup__less__iff,axiom,
! [X4: set_nat,A: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_nat @ X3 @ A ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1159_inj__on__iff__surj,axiom,
! [A2: set_nat,A8: set_option_list_o] :
( ( A2 != bot_bot_set_nat )
=> ( ( ? [F2: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A2 )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G3: option_list_o > nat] :
( ( image_5401538521246155887_o_nat @ G3 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1160_inj__on__iff__surj,axiom,
! [A2: set_nat,A8: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat @ G3 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1161_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1162_map__add__comm,axiom,
! [M1: nat > option_list_o,M2: nat > option_list_o] :
( ( ( inf_inf_set_nat @ ( dom_nat_list_o @ M1 ) @ ( dom_nat_list_o @ M2 ) )
= bot_bot_set_nat )
=> ( ( map_add_nat_list_o @ M1 @ M2 )
= ( map_add_nat_list_o @ M2 @ M1 ) ) ) ).
% map_add_comm
thf(fact_1163_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1164_range__from__nat__into__subset,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( counta7321652538601044515to_nat @ A2 ) @ top_top_set_nat ) @ A2 ) ) ).
% range_from_nat_into_subset
thf(fact_1165_SUP__countable__SUP,axiom,
! [A2: set_nat,G: nat > extended_ereal] :
( ( A2 != bot_bot_set_nat )
=> ? [F5: nat > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F5 @ top_top_set_nat ) @ ( image_4309273772856505399_ereal @ G @ A2 ) )
& ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F5 @ top_top_set_nat ) ) ) ) ) ).
% SUP_countable_SUP
thf(fact_1166_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1167_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1168_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1169_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_1170_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1171_mono__Max__commute,axiom,
! [F: nat > nat,A2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_1172_Max_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_1173_Max__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_1174_Max__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_1175_Max__ge,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max_ge
thf(fact_1176_Max__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_1177_Max__eq__if,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( ord_less_eq_nat @ X2 @ Xa ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A2 )
& ( ord_less_eq_nat @ X2 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_1178_Max_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_1179_Max__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A2 )
= M )
= ( ( member_nat @ M @ A2 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1180_Max__ge__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1181_eq__Max__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ( member_nat @ M @ A2 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1182_Max_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A2 )
=> ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1183_Max_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1184_Max__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_1185_cSup__eq__Max,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X4 )
= ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% cSup_eq_Max
thf(fact_1186_bij__betwI_H,axiom,
! [X4: set_nat,F: nat > nat,Y4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ X4 )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( member_nat @ ( F @ X2 ) @ Y4 ) )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ Y4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ X4 )
& ( Y3
= ( F @ X6 ) ) ) )
=> ( bij_betw_nat_nat @ F @ X4 @ Y4 ) ) ) ) ).
% bij_betwI'
thf(fact_1187_Max__mono,axiom,
! [M8: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M8 @ N )
=> ( ( M8 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M8 ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% Max_mono
thf(fact_1188_Max_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 @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ).
% Max.subset_imp
thf(fact_1189_card__le__Suc__Max,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1190_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X9: set_nat] : ( if_nat @ ( X9 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_1191_mono__Min__commute,axiom,
! [F: nat > nat,A2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_1192_lfp__Kleene__iter,axiom,
! [F: set_nat > set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K ) @ F @ bot_bot_set_nat )
= ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) )
=> ( ( comple7975543026063415949et_nat @ F )
= ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_1193_surj__fn,axiom,
! [F: nat > nat,N2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( compow_nat_nat @ N2 @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surj_fn
thf(fact_1194_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1195_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_1196_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_1197_mono__pow,axiom,
! [F: set_nat > set_nat,N2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N2 @ F ) ) ) ).
% mono_pow
thf(fact_1198_funpow__mono,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat,N2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N2 @ F @ A2 ) @ ( compow8708494347934031032et_nat @ N2 @ F @ B ) ) ) ) ).
% funpow_mono
thf(fact_1199_funpow__mono,axiom,
! [F: nat > nat,A2: nat,B: nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ N2 @ F @ A2 ) @ ( compow_nat_nat @ N2 @ F @ B ) ) ) ) ).
% funpow_mono
thf(fact_1200_bij__betw__funpow,axiom,
! [F: nat > nat,S: set_nat,N2: nat] :
( ( bij_betw_nat_nat @ F @ S @ S )
=> ( bij_betw_nat_nat @ ( compow_nat_nat @ N2 @ F ) @ S @ S ) ) ).
% bij_betw_funpow
thf(fact_1201_bij__fn,axiom,
! [F: nat > nat,N2: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( compow_nat_nat @ N2 @ F ) @ top_top_set_nat @ top_top_set_nat ) ) ).
% bij_fn
thf(fact_1202_inj__fn,axiom,
! [F: nat > nat,N2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( compow_nat_nat @ N2 @ F ) @ top_top_set_nat ) ) ).
% inj_fn
thf(fact_1203_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1204_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ord_less_eq_nat @ X @ Y3 ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1205_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_1206_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1207_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A2 )
=> ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1208_eq__Min__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat @ M @ A2 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1209_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1210_Min__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M )
= ( ( member_nat @ M @ A2 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1211_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_1212_Kleene__iter__gpfp,axiom,
! [F: set_nat > set_nat,P5: set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ P5 @ ( F @ P5 ) )
=> ( ord_less_eq_set_nat @ P5 @ ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_1213_Kleene__iter__lpfp,axiom,
! [F: set_nat > set_nat,P5: set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ ( F @ P5 ) @ P5 )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) @ P5 ) ) ) ).
% Kleene_iter_lpfp
thf(fact_1214_Kleene__iter__lpfp,axiom,
! [F: nat > nat,P5: nat,K: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ P5 ) @ P5 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ K @ F @ bot_bot_nat ) @ P5 ) ) ) ).
% Kleene_iter_lpfp
thf(fact_1215_funpow__mono2,axiom,
! [F: set_nat > set_nat,I: nat,J2: nat,X: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ ( F @ X ) )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ I @ F @ X ) @ ( compow8708494347934031032et_nat @ J2 @ F @ Y ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_1216_funpow__mono2,axiom,
! [F: nat > nat,I: nat,J2: nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ ( F @ X ) )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ I @ F @ X ) @ ( compow_nat_nat @ J2 @ F @ Y ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_1217_Min_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 @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1218_Min__antimono,axiom,
! [M8: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M8 @ N )
=> ( ( M8 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M8 ) ) ) ) ) ).
% Min_antimono
thf(fact_1219_funpow__increasing,axiom,
! [M: nat,N2: nat,F: set_nat > set_nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N2 @ F @ top_top_set_nat ) @ ( compow8708494347934031032et_nat @ M @ F @ top_top_set_nat ) ) ) ) ).
% funpow_increasing
thf(fact_1220_funpow__decreasing,axiom,
! [M: nat,N2: nat,F: set_nat > set_nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ M @ F @ bot_bot_set_nat ) @ ( compow8708494347934031032et_nat @ N2 @ F @ bot_bot_set_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_1221_funpow__decreasing,axiom,
! [M: nat,N2: nat,F: nat > nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ M @ F @ bot_bot_nat ) @ ( compow_nat_nat @ N2 @ F @ bot_bot_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_1222_lfp__funpow,axiom,
! [F: set_nat > set_nat,N2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( comple7975543026063415949et_nat @ ( compow8708494347934031032et_nat @ ( suc @ N2 ) @ F ) )
= ( comple7975543026063415949et_nat @ F ) ) ) ).
% lfp_funpow
thf(fact_1223_gfp__Kleene__iter,axiom,
! [F: set_nat > set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K ) @ F @ top_top_set_nat )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) )
=> ( ( comple1596078789208929544et_nat @ F )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_1224_Compl__anti__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1225_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1226_finite__compl,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1227_Compl__disjoint,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_1228_Compl__disjoint2,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_1229_surj__Compl__image__subset,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1230_disjoint__eq__subset__Compl,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1231_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1232_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1233_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1234_weak__coinduct__image,axiom,
! [A: nat,X4: set_nat,G: nat > nat,F: set_nat > set_nat] :
( ( member_nat @ A @ X4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ X4 ) @ ( F @ ( image_nat_nat @ G @ X4 ) ) )
=> ( member_nat @ ( G @ A ) @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1235_gfp__least,axiom,
! [F: set_nat > set_nat,X4: set_nat] :
( ! [U3: set_nat] :
( ( ord_less_eq_set_nat @ U3 @ ( F @ U3 ) )
=> ( ord_less_eq_set_nat @ U3 @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple1596078789208929544et_nat @ F ) @ X4 ) ) ).
% gfp_least
thf(fact_1236_gfp__upperbound,axiom,
! [X4: set_nat,F: set_nat > set_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( F @ X4 ) )
=> ( ord_less_eq_set_nat @ X4 @ ( comple1596078789208929544et_nat @ F ) ) ) ).
% gfp_upperbound
thf(fact_1237_gfp__mono,axiom,
! [F: set_nat > set_nat,G: set_nat > set_nat] :
( ! [Z6: set_nat] : ( ord_less_eq_set_nat @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_less_eq_set_nat @ ( comple1596078789208929544et_nat @ F ) @ ( comple1596078789208929544et_nat @ G ) ) ) ).
% gfp_mono
thf(fact_1238_weak__coinduct,axiom,
! [A: nat,X4: set_nat,F: set_nat > set_nat] :
( ( member_nat @ A @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ ( F @ X4 ) )
=> ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct
thf(fact_1239_bij__image__Compl__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1240_gfp__eqI,axiom,
! [F4: set_nat > set_nat,X: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z4: set_nat] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq_set_nat @ Z4 @ X ) )
=> ( ( comple1596078789208929544et_nat @ F4 )
= X ) ) ) ) ).
% gfp_eqI
thf(fact_1241_gfp__unfold,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( comple1596078789208929544et_nat @ F )
= ( F @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% gfp_unfold
thf(fact_1242_gfp__fixpoint,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( F @ ( comple1596078789208929544et_nat @ F ) )
= ( comple1596078789208929544et_nat @ F ) ) ) ).
% gfp_fixpoint
thf(fact_1243_def__gfp__unfold,axiom,
! [A2: set_nat,F: set_nat > set_nat] :
( ( A2
= ( comple1596078789208929544et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( A2
= ( F @ A2 ) ) ) ) ).
% def_gfp_unfold
thf(fact_1244_inj__image__Compl__subset,axiom,
! [F: nat > option_list_o,A2: set_nat] :
( ( inj_on1630180835328728801list_o @ F @ top_top_set_nat )
=> ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus2228965239982383419list_o @ ( image_4575287668734308173list_o @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1245_inj__image__Compl__subset,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1246_lfp__le__gfp,axiom,
! [F: set_nat > set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ ( comple1596078789208929544et_nat @ F ) ) ) ).
% lfp_le_gfp
thf(fact_1247_gfp__funpow,axiom,
! [F: set_nat > set_nat,N2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( comple1596078789208929544et_nat @ ( compow8708494347934031032et_nat @ ( suc @ N2 ) @ F ) )
= ( comple1596078789208929544et_nat @ F ) ) ) ).
% gfp_funpow
thf(fact_1248_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1249_ComplI,axiom,
! [C2: nat,A2: set_nat] :
( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_1250_Compl__iff,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_1251_ComplD,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_1252_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_1253_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1254_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1255_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1256_sum__le__card__Max,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( times_times_nat @ ( finite_card_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_1257_graph__map__add,axiom,
! [M1: nat > option_list_o,M2: nat > option_list_o] :
( ( ( inf_inf_set_nat @ ( dom_nat_list_o @ M1 ) @ ( dom_nat_list_o @ M2 ) )
= bot_bot_set_nat )
=> ( ( graph_nat_list_o @ ( map_add_nat_list_o @ M1 @ M2 ) )
= ( sup_su4620230350752640421list_o @ ( graph_nat_list_o @ M1 ) @ ( graph_nat_list_o @ M2 ) ) ) ) ).
% graph_map_add
thf(fact_1258_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_1259_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1260_sup_Obounded__iff,axiom,
! [B4: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_nat @ B4 @ A )
& ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1261_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1262_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X @ Z3 )
& ( ord_less_eq_set_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1263_le__sup__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1264_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1265_finite__Un,axiom,
! [F4: set_nat,G5: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G5 ) )
= ( ( finite_finite_nat @ F4 )
& ( finite_finite_nat @ G5 ) ) ) ).
% finite_Un
thf(fact_1266_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A2 @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( dom_nat_list_o @ ( prefix6319276831915272717e_Nb_e @ l ) )
= ( set_ord_lessThan_nat @ l ) ) ).
%------------------------------------------------------------------------------