TPTP Problem File: SLH0226^1.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 : Fishers_Inequality/0034_Incidence_Matrices/prob_00072_003381__27969142_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1519 ( 542 unt; 239 typ; 0 def)
% Number of atoms : 3697 (1302 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10206 ( 377 ~; 29 |; 228 &;7965 @)
% ( 0 <=>;1607 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 788 ( 788 >; 0 *; 0 +; 0 <<)
% Number of symbols : 208 ( 205 usr; 41 con; 0-4 aty)
% Number of variables : 3146 ( 143 ^;2912 !; 91 ?;3146 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:46:14.969
%------------------------------------------------------------------------------
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
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thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001tf__a,type,
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thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001tf__a,type,
neg_nu6917059380386235053_inc_a: a > a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_Mt__Nat__Onat_J,type,
ord_less_eq_a_nat: ( a > nat ) > ( a > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_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_001_062_It__Product____Type__Ounit_M_Eo_J,type,
top_to2465898995584390880unit_o: product_unit > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
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__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
top_to2690860209552263555t_unit: set_op3165557761946182707t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to8544742955230171288t_unit: set_Pr4334478416066269672t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
top_to2612598781856825737_nat_a: set_Pr4193341848836149977_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to5974110478112770290it_nat: set_Pr1763845938948868674it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to1835807148980544151t_unit: set_Pr5094982260447487303t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__a_J_J,type,
top_to1216281454841048712unit_a: set_Pr1310170126721327416unit_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
top_to3353692345378799459_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J_J,type,
top_to6636102223169616742t_unit: set_Pr6729990895049895446t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to8063371432257647191od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
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__Set__Oset_It__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
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_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to5465250082899874788t_unit: set_Su7539578257924484756t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
top_to54524901450547413_nat_a: set_Sum_sum_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to2894617605782473790it_nat: set_Su4968945780807083758it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to2771918933716375115t_unit: set_Su4110612849109743515t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mtf__a_J_J,type,
top_to5559247480540603964unit_a: set_Su350143336570148748unit_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Product____Type__Ounit_J_J,type,
top_to1755696212014396186t_unit: set_Su5769964104898716778t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
top_to8848906000605539851um_a_a: set_Sum_sum_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Power__Products_Odeg__fun_001t__Nat__Onat_001t__Nat__Onat,type,
power_2594616646957108920at_nat: ( nat > nat ) > nat ).
thf(sy_c_Power__Products_Odeg__fun_001tf__a_001t__Nat__Onat,type,
power_deg_fun_a_nat: ( a > nat ) > nat ).
thf(sy_c_Power__Products_Osupp__fun_001t__Nat__Onat_001t__Nat__Onat,type,
power_3957698682672687778at_nat: ( nat > nat ) > set_nat ).
thf(sy_c_Power__Products_Osupp__fun_001t__Nat__Onat_001tf__a,type,
power_supp_fun_nat_a: ( nat > a ) > set_nat ).
thf(sy_c_Power__Products_Osupp__fun_001tf__a_001t__Nat__Onat,type,
power_supp_fun_a_nat: ( a > nat ) > set_a ).
thf(sy_c_Power__Products_Osupp__fun_001tf__a_001tf__a,type,
power_supp_fun_a_a: ( a > a ) > set_a ).
thf(sy_c_Ring__Hom_Oone__hom_001t__Nat__Onat_001t__Nat__Onat,type,
ring_one_hom_nat_nat: ( nat > nat ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom_001t__Nat__Onat_001tf__a,type,
ring_one_hom_nat_a: ( nat > a ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom_001tf__a_001t__Nat__Onat,type,
ring_one_hom_a_nat: ( a > nat ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom_001tf__a_001tf__a,type,
ring_one_hom_a_a: ( a > a ) > $o ).
thf(sy_c_Ring__Hom_Ozero__hom_001t__Nat__Onat_001t__Nat__Onat,type,
ring_z4445335182927245238at_nat: ( nat > nat ) > $o ).
thf(sy_c_Ring__Hom_Ozero__hom_001t__Nat__Onat_001tf__a,type,
ring_zero_hom_nat_a: ( nat > a ) > $o ).
thf(sy_c_Ring__Hom_Ozero__hom_001tf__a_001t__Nat__Onat,type,
ring_zero_hom_a_nat: ( a > nat ) > $o ).
thf(sy_c_Ring__Hom_Ozero__hom_001tf__a_001tf__a,type,
ring_zero_hom_a_a: ( a > a ) > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Product____Type__Ounit,type,
pow_Product_unit: set_Product_unit > set_set_Product_unit ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001tf__a,type,
image_Product_unit_a: ( product_unit > a ) > set_Product_unit > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Ounit,type,
image_a_Product_unit: ( a > product_unit ) > set_a > set_Product_unit ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Ounit,type,
is_sin2160648248035936513t_unit: set_Product_unit > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
vimage4884490618288580032t_unit: ( nat > product_unit ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
vimage6253328473476588386it_nat: ( product_unit > nat ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
vimage7995052115951654139t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001tf__a,type,
vimage4490873842868098028unit_a: ( product_unit > a ) > set_a > set_Product_unit ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Product____Type__Ounit,type,
vimage3195509354877090634t_unit: ( a > product_unit ) > set_Product_unit > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Bs,type,
bs: list_set_b ).
thf(sy_v_Vs,type,
vs: list_b ).
% Relevant facts (1274)
thf(fact_0_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_1_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_2_inc__mat__one__zero__elems,axiom,
! [Vs: list_b,Bs: list_set_b] : ( ord_less_eq_set_a @ ( elements_mat_a @ ( incide3343782147941204138of_b_a @ Vs @ Bs ) ) @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) ) ).
% inc_mat_one_zero_elems
thf(fact_3_infinite__imp__elem,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ? [X: a] : ( member_a @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_4_infinite__imp__elem,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ? [X: nat] : ( member_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_5_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_6_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_7_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_8_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_9_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_10_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_11_finite__transitivity__chain,axiom,
! [A: set_a,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X: a] :
~ ( R @ X @ X )
=> ( ! [X: a,Y: a,Z: a] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X @ Z ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Y2: a] :
( ( member_a @ Y2 @ A )
& ( R @ X @ Y2 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_12_finite__transitivity__chain,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
~ ( R @ X @ X )
=> ( ! [X: nat,Y: nat,Z: nat] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X @ Z ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ A )
& ( R @ X @ Y2 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_13_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_14_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_15_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_16_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_17_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_18_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_19_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ A2 @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_20_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_21_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_22_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_23_finite__subset__induct_H,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_24_finite__subset__induct_H,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_25_finite__subset__induct,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_26_finite__subset__induct,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_27_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_28_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_29_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_30_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_31_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_32_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_33_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A5: set_a] :
( ( A5 = bot_bot_set_a )
| ? [A6: set_a,B2: a] :
( ( A5
= ( insert_a @ B2 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_34_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A6: set_nat,B2: nat] :
( ( A5
= ( insert_nat @ B2 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_35_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A3: a] :
( A2
= ( insert_a @ A3 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_36_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A4: set_nat] :
( ? [A3: nat] :
( A2
= ( insert_nat @ A3 @ A4 ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_37_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_38_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_39_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_40_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_41_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_42_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_43_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_44_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_45_singleton__insert__inj__eq,axiom,
! [B3: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B3 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_46_singleton__insert__inj__eq,axiom,
! [B3: a,A2: a,A: set_a] :
( ( ( insert_a @ B3 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_47_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B3: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_48_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B3: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B3 @ bot_bot_set_a ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_49_insert__subset,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_50_insert__subset,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_51_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_52_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_53_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_54_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_55_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_56_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_57_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_58_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 ) @ ( F3 @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_59_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 ) @ ( F3 @ X ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_60_subset__singletonD,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_61_subset__singletonD,axiom,
! [A: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_62_subset__singleton__iff,axiom,
! [X3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
| ( X3
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_63_subset__singleton__iff,axiom,
! [X3: set_a,A2: a] :
( ( ord_less_eq_set_a @ X3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X3 = bot_bot_set_a )
| ( X3
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_64_ex__min,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ord_less_eq_nat @ X @ Xa ) ) ) ) ) ).
% ex_min
thf(fact_65_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_66_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_67_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_68_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_70_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_71_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_72_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_73_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_74_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_75_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_76_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_77_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_78_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_82_insert__absorb2,axiom,
! [X2: a,A: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A ) )
= ( insert_a @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_83_insert__absorb2,axiom,
! [X2: nat,A: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A ) )
= ( insert_nat @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_84_insert__iff,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_85_insert__iff,axiom,
! [A2: a,B3: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_86_insertCI,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_87_insertCI,axiom,
! [A2: a,B: set_a,B3: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_88_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_89_zero__reorient,axiom,
! [X2: a] :
( ( zero_zero_a = X2 )
= ( X2 = zero_zero_a ) ) ).
% zero_reorient
thf(fact_90_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_91_one__reorient,axiom,
! [X2: a] :
( ( one_one_a = X2 )
= ( X2 = one_one_a ) ) ).
% one_reorient
thf(fact_92_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_93_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_94_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_95_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_96_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_97_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_98_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_99_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_100_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_101_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_102_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A6: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_103_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_104_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_105_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_106_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_107_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_108_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_109_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_110_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_111_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_112_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_113_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_114_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_115_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_116_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_117_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_118_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_119_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_120_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_121_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_122_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_123_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_124_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_125_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_126_insert__commute,axiom,
! [X2: a,Y4: a,A: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y4 @ A ) )
= ( insert_a @ Y4 @ ( insert_a @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_127_insert__commute,axiom,
! [X2: nat,Y4: nat,A: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y4 @ A ) )
= ( insert_nat @ Y4 @ ( insert_nat @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_128_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B3: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B3 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B3 @ C3 ) )
& ~ ( member_nat @ B3 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_129_insert__eq__iff,axiom,
! [A2: a,A: set_a,B3: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B3 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B3 @ C3 ) )
& ~ ( member_a @ B3 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_130_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_131_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_132_insert__ident,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ A )
= ( insert_nat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_133_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A )
= ( insert_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_134_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat @ X2 @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X2 @ B5 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_135_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a @ X2 @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_136_insertI2,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_137_insertI2,axiom,
! [A2: a,B: set_a,B3: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_138_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_139_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_140_insertE,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_141_insertE,axiom,
! [A2: a,B3: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_142_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_143_zero__min,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_min
thf(fact_144_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_145_zero__neq__one,axiom,
zero_zero_a != one_one_a ).
% zero_neq_one
thf(fact_146_singleton__inject,axiom,
! [A2: a,B3: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B3 @ bot_bot_set_a ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_147_singleton__inject,axiom,
! [A2: nat,B3: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_148_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_149_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_150_doubleton__eq__iff,axiom,
! [A2: a,B3: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_151_doubleton__eq__iff,axiom,
! [A2: nat,B3: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_152_singleton__iff,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_153_singleton__iff,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_154_singletonD,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_155_singletonD,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_156_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_157_subset__insertI2,axiom,
! [A: set_a,B: set_a,B3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_158_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_159_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_160_subset__insert,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_161_subset__insert,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_162_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_163_insert__mono,axiom,
! [C2: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_164_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_165_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_166_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_167_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_168_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_169_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_170_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_171_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_172_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_173_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_174_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_175_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_176_arg__min__least,axiom,
! [S: set_a,Y4: a,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y4 @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic6340287419671400565_a_nat @ F3 @ S ) ) @ ( F3 @ Y4 ) ) ) ) ) ).
% arg_min_least
thf(fact_177_arg__min__least,axiom,
! [S: set_nat,Y4: nat,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y4 @ S )
=> ( ord_less_eq_nat @ ( F3 @ ( lattic7446932960582359483at_nat @ F3 @ S ) ) @ ( F3 @ Y4 ) ) ) ) ) ).
% arg_min_least
thf(fact_178_insert__subsetI,axiom,
! [X2: nat,A: set_nat,X3: set_nat] :
( ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_179_insert__subsetI,axiom,
! [X2: a,A: set_a,X3: set_a] :
( ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_180_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_181_subset__emptyI,axiom,
! [A: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_182_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_183_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_184_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_185_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_186_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_187_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_188_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_189_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X: a,Y: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( X = Y ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_190_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_191_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_192_le__cases3,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_193_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_194_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_197_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_198_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_200_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_201_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_202_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_203_order__antisym,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_204_order__antisym,axiom,
! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_205_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_206_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_207_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_208_order__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_209_order__trans,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_210_order__trans,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_211_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( P @ A3 @ B6 ) )
=> ( ! [A3: nat,B6: nat] :
( ( P @ B6 @ A3 )
=> ( P @ A3 @ B6 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_212_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ( ord_less_eq_nat @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_213_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A5: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_214_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_215_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_216_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_217_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_218_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_219_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_220_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_221_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_222_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_223_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_224_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ( ord_less_eq_nat @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_225_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A5: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_226_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_227_order__subst1,axiom,
! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_228_order__subst1,axiom,
! [A2: nat,F3: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_229_order__subst1,axiom,
! [A2: nat,F3: set_a > nat,B3: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_230_order__subst1,axiom,
! [A2: set_nat,F3: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_231_order__subst1,axiom,
! [A2: set_nat,F3: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_232_order__subst1,axiom,
! [A2: set_nat,F3: set_a > set_nat,B3: set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_233_order__subst1,axiom,
! [A2: set_a,F3: nat > set_a,B3: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_234_order__subst1,axiom,
! [A2: set_a,F3: set_nat > set_a,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_235_order__subst1,axiom,
! [A2: set_a,F3: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_236_order__subst2,axiom,
! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_237_order__subst2,axiom,
! [A2: nat,B3: nat,F3: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_238_order__subst2,axiom,
! [A2: nat,B3: nat,F3: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B3 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_239_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_240_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_241_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_242_order__subst2,axiom,
! [A2: set_a,B3: set_a,F3: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_243_order__subst2,axiom,
! [A2: set_a,B3: set_a,F3: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_244_order__subst2,axiom,
! [A2: set_a,B3: set_a,F3: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B3 ) @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_245_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_246_order__eq__refl,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_247_order__eq__refl,axiom,
! [X2: set_a,Y4: set_a] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_a @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_248_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_249_ord__eq__le__subst,axiom,
! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_250_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: nat > set_nat,B3: nat,C: nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_251_ord__eq__le__subst,axiom,
! [A2: set_a,F3: nat > set_a,B3: nat,C: nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A2: nat,F3: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A2: set_a,F3: set_nat > set_a,B3: set_nat,C: set_nat] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A2: nat,F3: set_a > nat,B3: set_a,C: set_a] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__eq__le__subst,axiom,
! [A2: set_nat,F3: set_a > set_nat,B3: set_a,C: set_a] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
! [A2: set_a,F3: set_a > set_a,B3: set_a,C: set_a] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_258_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F3: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_260_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F3: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_261_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_262_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_263_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F3: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_264_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F3: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_265_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F3: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_266_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F3: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C )
=> ( ! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_267_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_268_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_269_order__antisym__conv,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_270_order__antisym__conv,axiom,
! [Y4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_271_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X: a] :
( A
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_272_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X: nat] :
( A
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_273_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X4: a] :
( A6
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_274_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X4: nat] :
( A6
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_275_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_276_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_277_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_278_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_279_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_280_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_281_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_282_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_283_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_284_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A6: set_nat] : ( A6 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_285_Inf__fin_Osubset__imp,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_286_Inf__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_287_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_288_Sup__fin_Osubset__imp,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_289_Sup__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_290_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_291_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: a > nat,X2: a] :
( ( matrix4679759516020283290_a_nat @ Hom )
=> ( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( Hom @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_a ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_292_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: a > a,X2: a] :
( ( matrix354569784383077812om_a_a @ Hom )
=> ( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( Hom @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_a ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_293_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > nat,X2: nat] :
( ( matrix3195545623030104918at_nat @ Hom )
=> ( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_294_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > a,X2: nat] :
( ( matrix8711691017198569720_nat_a @ Hom )
=> ( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_295_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: a > nat,X2: a] :
( ( matrix4679759516020283290_a_nat @ Hom )
=> ( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( Hom @ X2 )
= one_one_nat )
= ( X2 = one_one_a ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_296_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: a > a,X2: a] :
( ( matrix354569784383077812om_a_a @ Hom )
=> ( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( Hom @ X2 )
= one_one_a )
= ( X2 = one_one_a ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_297_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: nat > nat,X2: nat] :
( ( matrix3195545623030104918at_nat @ Hom )
=> ( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X2 )
= one_one_nat )
= ( X2 = one_one_nat ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_298_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: nat > a,X2: nat] :
( ( matrix8711691017198569720_nat_a @ Hom )
=> ( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X2 )
= one_one_a )
= ( X2 = one_one_nat ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_299_Sup__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_300_Inf__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_301_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_302_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_303_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_304_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_305_Inf__fin_OcoboundedI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_306_Inf__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_307_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_308_Sup__fin_OcoboundedI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ A2 @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_309_Sup__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_310_Inf__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_311_Inf__fin_OboundedE,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A ) )
=> ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A )
=> ( ord_less_eq_set_nat @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_312_Inf__fin_OboundedE,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) )
=> ! [A7: set_a] :
( ( member_set_a @ A7 @ A )
=> ( ord_less_eq_set_a @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_313_Inf__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ X2 @ A3 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_314_Inf__fin_OboundedI,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
=> ( ord_less_eq_set_nat @ X2 @ A3 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_315_Inf__fin_OboundedI,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ X2 @ A3 ) )
=> ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_316_Sup__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_317_Sup__fin_OboundedE,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X2 )
=> ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A )
=> ( ord_less_eq_set_nat @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_318_Sup__fin_OboundedE,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 )
=> ! [A7: set_a] :
( ( member_set_a @ A7 @ A )
=> ( ord_less_eq_set_a @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_319_Sup__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ A3 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_320_Sup__fin_OboundedI,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
=> ( ord_less_eq_set_nat @ A3 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_321_Sup__fin_OboundedI,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ A3 @ X2 ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_322_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ X2 @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_323_Inf__fin_Obounded__iff,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X2 @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_324_Inf__fin_Obounded__iff,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X2 @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_325_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_326_Sup__fin_Obounded__iff,axiom,
! [A: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X2 )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_327_Sup__fin_Obounded__iff,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_328_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_329_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_330_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_331_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_332_dbl__inc__simps_I2_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ zero_zero_a )
= one_one_a ) ).
% dbl_inc_simps(2)
thf(fact_333_of__zero__hom_Ohom__zero,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_334_of__zero__hom_Ohom__zero,axiom,
( ( matrix8283685725398817568_nat_a @ zero_zero_nat )
= zero_zero_a ) ).
% of_zero_hom.hom_zero
thf(fact_335_of__zero__hom_Ohom__zero,axiom,
( ( matrix4251754224220531138_a_nat @ zero_zero_a )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_336_of__zero__hom_Ohom__zero,axiom,
( ( matrix7568498694042281356ne_a_a @ zero_zero_a )
= zero_zero_a ) ).
% of_zero_hom.hom_zero
thf(fact_337_of__zero__hom_Ohom__0__iff,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_338_of__zero__hom_Ohom__0__iff,axiom,
! [X2: a] :
( ( ( matrix4251754224220531138_a_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_a ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_339_of__zero__hom_Ohom__0__iff,axiom,
! [X2: nat] :
( ( ( matrix8283685725398817568_nat_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_340_of__zero__hom_Ohom__0__iff,axiom,
! [X2: a] :
( ( ( matrix7568498694042281356ne_a_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_a ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_341_of__zero__neq__one__0,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_342_of__zero__neq__one__0,axiom,
( ( matrix8283685725398817568_nat_a @ zero_zero_nat )
= zero_zero_a ) ).
% of_zero_neq_one_0
thf(fact_343_of__zero__neq__one__0,axiom,
( ( matrix4251754224220531138_a_nat @ zero_zero_a )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_344_of__zero__neq__one__0,axiom,
( ( matrix7568498694042281356ne_a_a @ zero_zero_a )
= zero_zero_a ) ).
% of_zero_neq_one_0
thf(fact_345_of__zero__neq__one__0__iff,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_346_of__zero__neq__one__0__iff,axiom,
! [X2: a] :
( ( ( matrix4251754224220531138_a_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_a ) ) ).
% of_zero_neq_one_0_iff
thf(fact_347_of__zero__neq__one__0__iff,axiom,
! [X2: nat] :
( ( ( matrix8283685725398817568_nat_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_348_of__zero__neq__one__0__iff,axiom,
! [X2: a] :
( ( ( matrix7568498694042281356ne_a_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_a ) ) ).
% of_zero_neq_one_0_iff
thf(fact_349_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_350_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix8283685725398817568_nat_a @ one_one_nat )
= one_one_a ) ).
% of_inj_on_01_hom.hom_one
thf(fact_351_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4251754224220531138_a_nat @ one_one_a )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_352_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix7568498694042281356ne_a_a @ one_one_a )
= one_one_a ) ).
% of_inj_on_01_hom.hom_one
thf(fact_353_of__zero__neq__one__1,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_354_of__zero__neq__one__1,axiom,
( ( matrix8283685725398817568_nat_a @ one_one_nat )
= one_one_a ) ).
% of_zero_neq_one_1
thf(fact_355_of__zero__neq__one__1,axiom,
( ( matrix4251754224220531138_a_nat @ one_one_a )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_356_of__zero__neq__one__1,axiom,
( ( matrix7568498694042281356ne_a_a @ one_one_a )
= one_one_a ) ).
% of_zero_neq_one_1
thf(fact_357_of__zero__hom_Ohom__0,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_358_of__zero__hom_Ohom__0,axiom,
! [X2: a] :
( ( ( matrix4251754224220531138_a_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_a ) ) ).
% of_zero_hom.hom_0
thf(fact_359_of__zero__hom_Ohom__0,axiom,
! [X2: nat] :
( ( ( matrix8283685725398817568_nat_a @ X2 )
= zero_zero_a )
=> ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_360_of__zero__hom_Ohom__0,axiom,
! [X2: a] :
( ( ( matrix7568498694042281356ne_a_a @ X2 )
= zero_zero_a )
=> ( X2 = zero_zero_a ) ) ).
% of_zero_hom.hom_0
thf(fact_361_of__zero__neq__one__def,axiom,
( matrix700445748609480494at_nat
= ( ^ [X4: nat] : ( if_nat @ ( X4 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_362_of__zero__neq__one__def,axiom,
( matrix8283685725398817568_nat_a
= ( ^ [X4: nat] : ( if_a @ ( X4 = zero_zero_nat ) @ zero_zero_a @ one_one_a ) ) ) ).
% of_zero_neq_one_def
thf(fact_363_of__zero__neq__one__def,axiom,
( matrix4251754224220531138_a_nat
= ( ^ [X4: a] : ( if_nat @ ( X4 = zero_zero_a ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_364_of__zero__neq__one__def,axiom,
( matrix7568498694042281356ne_a_a
= ( ^ [X4: a] : ( if_a @ ( X4 = zero_zero_a ) @ zero_zero_a @ one_one_a ) ) ) ).
% of_zero_neq_one_def
thf(fact_365_of__inj__on__01__hom_Oinj__0__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( matrix4251754224220531138_a_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_a ) ) ) ).
% of_inj_on_01_hom.inj_0_iff
thf(fact_366_of__inj__on__01__hom_Oinj__0__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( matrix7568498694042281356ne_a_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_a ) ) ) ).
% of_inj_on_01_hom.inj_0_iff
thf(fact_367_of__inj__on__01__hom_Oinj__0__iff,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ) ).
% of_inj_on_01_hom.inj_0_iff
thf(fact_368_of__inj__on__01__hom_Oinj__0__iff,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( matrix8283685725398817568_nat_a @ X2 )
= zero_zero_a )
= ( X2 = zero_zero_nat ) ) ) ).
% of_inj_on_01_hom.inj_0_iff
thf(fact_369_of__inj__on__01__hom_Oinj__1__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( matrix4251754224220531138_a_nat @ X2 )
= one_one_nat )
= ( X2 = one_one_a ) ) ) ).
% of_inj_on_01_hom.inj_1_iff
thf(fact_370_of__inj__on__01__hom_Oinj__1__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( insert_a @ zero_zero_a @ ( insert_a @ one_one_a @ bot_bot_set_a ) ) )
=> ( ( ( matrix7568498694042281356ne_a_a @ X2 )
= one_one_a )
= ( X2 = one_one_a ) ) ) ).
% of_inj_on_01_hom.inj_1_iff
thf(fact_371_of__inj__on__01__hom_Oinj__1__iff,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( matrix700445748609480494at_nat @ X2 )
= one_one_nat )
= ( X2 = one_one_nat ) ) ) ).
% of_inj_on_01_hom.inj_1_iff
thf(fact_372_of__inj__on__01__hom_Oinj__1__iff,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( matrix8283685725398817568_nat_a @ X2 )
= one_one_a )
= ( X2 = one_one_nat ) ) ) ).
% of_inj_on_01_hom.inj_1_iff
thf(fact_373_deg__fun__eq__0__iff,axiom,
! [S3: a > nat] :
( ( finite_finite_a @ ( power_supp_fun_a_nat @ S3 ) )
=> ( ( ( power_deg_fun_a_nat @ S3 )
= zero_zero_nat )
= ( S3 = zero_zero_a_nat ) ) ) ).
% deg_fun_eq_0_iff
thf(fact_374_deg__fun__eq__0__iff,axiom,
! [S3: nat > nat] :
( ( finite_finite_nat @ ( power_3957698682672687778at_nat @ S3 ) )
=> ( ( ( power_2594616646957108920at_nat @ S3 )
= zero_zero_nat )
= ( S3 = zero_zero_nat_nat ) ) ) ).
% deg_fun_eq_0_iff
thf(fact_375_deg__fun__leq,axiom,
! [S3: a > nat,T3: a > nat] :
( ( finite_finite_a @ ( power_supp_fun_a_nat @ S3 ) )
=> ( ( finite_finite_a @ ( power_supp_fun_a_nat @ T3 ) )
=> ( ( ord_less_eq_a_nat @ S3 @ T3 )
=> ( ord_less_eq_nat @ ( power_deg_fun_a_nat @ S3 ) @ ( power_deg_fun_a_nat @ T3 ) ) ) ) ) ).
% deg_fun_leq
thf(fact_376_deg__fun__leq,axiom,
! [S3: nat > nat,T3: nat > nat] :
( ( finite_finite_nat @ ( power_3957698682672687778at_nat @ S3 ) )
=> ( ( finite_finite_nat @ ( power_3957698682672687778at_nat @ T3 ) )
=> ( ( ord_less_eq_nat_nat @ S3 @ T3 )
=> ( ord_less_eq_nat @ ( power_2594616646957108920at_nat @ S3 ) @ ( power_2594616646957108920at_nat @ T3 ) ) ) ) ) ).
% deg_fun_leq
thf(fact_377_Inf__fin_Oinsert,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_378_Sup__fin_Oinsert,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_379_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_380_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_381_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_382_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_383_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_384_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_385_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_386_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_387_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_388_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_389_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_390_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_391_Un__insert__left,axiom,
! [A2: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_392_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_393_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_394_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_395_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_396_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_397_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_398_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_399_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_400_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_401_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_402_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_403_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_404_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_405_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_406_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_407_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_408_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_409_disjoint__insert_I2_J,axiom,
! [A: set_a,B3: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B3 @ B ) ) )
= ( ~ ( member_a @ B3 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_410_disjoint__insert_I2_J,axiom,
! [A: set_nat,B3: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B3 @ B ) ) )
= ( ~ ( member_nat @ B3 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_411_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_412_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_413_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_414_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_415_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_416_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_417_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_418_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_419_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A6 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_420_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( sup_sup_set_a @ A6 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_421_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A8: set_nat] :
( ( ord_less_eq_set_nat @ A8 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_422_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_a @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_423_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_424_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_425_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_426_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_427_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_428_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_429_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_430_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_431_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_432_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_433_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_434_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_435_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_436_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_437_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_438_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_439_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_440_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_441_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_442_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ B )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_443_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_444_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_445_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_446_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_447_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_448_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_449_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_450_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_451_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_452_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_453_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_454_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_455_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_456_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_457_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_458_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_459_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_460_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_461_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_462_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_463_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_464_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_465_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_466_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_467_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_468_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_469_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_470_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_471_singleton__Un__iff,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ( insert_a @ X2 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_472_singleton__Un__iff,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_473_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_474_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_475_insert__is__Un,axiom,
( insert_a
= ( ^ [A5: a] : ( sup_sup_set_a @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_476_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_477_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_478_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_479_fun__eq__zeroI,axiom,
! [F3: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( power_3957698682672687778at_nat @ F3 ) )
=> ( ( F3 @ X )
= zero_zero_nat ) )
=> ( F3 = zero_zero_nat_nat ) ) ).
% fun_eq_zeroI
thf(fact_480_fun__eq__zeroI,axiom,
! [F3: a > nat] :
( ! [X: a] :
( ( member_a @ X @ ( power_supp_fun_a_nat @ F3 ) )
=> ( ( F3 @ X )
= zero_zero_nat ) )
=> ( F3 = zero_zero_a_nat ) ) ).
% fun_eq_zeroI
thf(fact_481_fun__eq__zeroI,axiom,
! [F3: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ ( power_supp_fun_nat_a @ F3 ) )
=> ( ( F3 @ X )
= zero_zero_a ) )
=> ( F3 = zero_zero_nat_a ) ) ).
% fun_eq_zeroI
thf(fact_482_fun__eq__zeroI,axiom,
! [F3: a > a] :
( ! [X: a] :
( ( member_a @ X @ ( power_supp_fun_a_a @ F3 ) )
=> ( ( F3 @ X )
= zero_zero_a ) )
=> ( F3 = zero_zero_a_a ) ) ).
% fun_eq_zeroI
thf(fact_483_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_484_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_485_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X2 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_486_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y ) @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_487_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X2 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_488_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y ) @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_489_sup__bot__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_490_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_491_sup__bot__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_492_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_493_bot__eq__sup__iff,axiom,
! [X2: set_a,Y4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_494_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_495_sup__eq__bot__iff,axiom,
! [X2: set_a,Y4: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y4 )
= bot_bot_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_496_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y4 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_497_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_498_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_499_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_500_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_501_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_502_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_503_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_504_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_505_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_506_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_507_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_508_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_509_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_510_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_511_inf_Obounded__iff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_512_inf_Obounded__iff,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B3 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_513_inf_Obounded__iff,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_514_le__inf__iff,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z3 ) )
= ( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_515_le__inf__iff,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y4 @ Z3 ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y4 )
& ( ord_less_eq_set_nat @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_516_le__inf__iff,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
= ( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_517_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_518_sup_Obounded__iff,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B3 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_519_sup_Obounded__iff,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_520_le__sup__iff,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y4 ) @ Z3 )
= ( ( ord_less_eq_nat @ X2 @ Z3 )
& ( ord_less_eq_nat @ Y4 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_521_le__sup__iff,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X2 @ Z3 )
& ( ord_less_eq_set_nat @ Y4 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_522_le__sup__iff,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ Z3 )
= ( ( ord_less_eq_set_a @ X2 @ Z3 )
& ( ord_less_eq_set_a @ Y4 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_523_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_524_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_525_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_526_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_527_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_528_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_529_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_530_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_531_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_532_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_533_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_534_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_535_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_536_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_537_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_538_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_539_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_540_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_541_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_542_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_543_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_544_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_545_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_546_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_547_inf__le1,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_le1
thf(fact_548_inf__le1,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ X2 ) ).
% inf_le1
thf(fact_549_inf__le1,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ X2 ) ).
% inf_le1
thf(fact_550_inf__le2,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_551_inf__le2,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_552_inf__le2,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_553_le__infE,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B3 ) ) ) ).
% le_infE
thf(fact_554_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B3 ) ) ) ).
% le_infE
thf(fact_555_le__infE,axiom,
! [X2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A2 )
=> ~ ( ord_less_eq_set_a @ X2 @ B3 ) ) ) ).
% le_infE
thf(fact_556_le__infI,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_557_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B3 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_558_le__infI,axiom,
! [X2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ B3 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_559_inf__mono,axiom,
! [A2: nat,C: nat,B3: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_560_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_561_inf__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_562_le__infI1,axiom,
! [A2: nat,X2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% le_infI1
thf(fact_563_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% le_infI1
thf(fact_564_le__infI1,axiom,
! [A2: set_a,X2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X2 ) ) ).
% le_infI1
thf(fact_565_le__infI2,axiom,
! [B3: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% le_infI2
thf(fact_566_le__infI2,axiom,
! [B3: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% le_infI2
thf(fact_567_le__infI2,axiom,
! [B3: set_a,X2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X2 ) ) ).
% le_infI2
thf(fact_568_inf_OorderE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2
= ( inf_inf_nat @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_569_inf_OorderE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_570_inf_OorderE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_571_inf_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_572_inf_OorderI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_573_inf_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_574_inf__unique,axiom,
! [F3: nat > nat > nat,X2: nat,Y4: nat] :
( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F3 @ X @ Y ) @ X )
=> ( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F3 @ X @ Y ) @ Y )
=> ( ! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( F3 @ Y @ Z ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_575_inf__unique,axiom,
! [F3: set_nat > set_nat > set_nat,X2: set_nat,Y4: set_nat] :
( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F3 @ X @ Y ) @ X )
=> ( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F3 @ X @ Y ) @ Y )
=> ( ! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( F3 @ Y @ Z ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_576_inf__unique,axiom,
! [F3: set_a > set_a > set_a,X2: set_a,Y4: set_a] :
( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F3 @ X @ Y ) @ X )
=> ( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F3 @ X @ Y ) @ Y )
=> ( ! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( F3 @ Y @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_577_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( inf_inf_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_578_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( inf_inf_set_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_579_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_580_inf_Oabsorb1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_581_inf_Oabsorb1,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_582_inf_Oabsorb1,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_583_inf_Oabsorb2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_584_inf_Oabsorb2,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_585_inf_Oabsorb2,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_586_inf__absorb1,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ).
% inf_absorb1
thf(fact_587_inf__absorb1,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( inf_inf_set_nat @ X2 @ Y4 )
= X2 ) ) ).
% inf_absorb1
thf(fact_588_inf__absorb1,axiom,
! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= X2 ) ) ).
% inf_absorb1
thf(fact_589_inf__absorb2,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_590_inf__absorb2,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_591_inf__absorb2,axiom,
! [Y4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_592_inf_OboundedE,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_593_inf_OboundedE,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_594_inf_OboundedE,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_595_inf_OboundedI,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_596_inf_OboundedI,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_597_inf_OboundedI,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_598_inf__greatest,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_599_inf__greatest,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y4 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_600_inf__greatest,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_601_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( A5
= ( inf_inf_nat @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_602_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_603_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
( A5
= ( inf_inf_set_a @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_604_inf_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_605_inf_Ocobounded1,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_606_inf_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_607_inf_Ocobounded2,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_608_inf_Ocobounded2,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_609_inf_Ocobounded2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_610_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( inf_inf_nat @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_611_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_612_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
( ( inf_inf_set_a @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_613_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_614_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_615_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A5: set_a] :
( ( inf_inf_set_a @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_616_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_617_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_618_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_619_inf_OcoboundedI2,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_620_inf_OcoboundedI2,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_621_inf_OcoboundedI2,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_622_inf__sup__ord_I4_J,axiom,
! [Y4: nat,X2: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_623_inf__sup__ord_I4_J,axiom,
! [Y4: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_624_inf__sup__ord_I4_J,axiom,
! [Y4: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_625_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_626_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_627_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_628_le__supE,axiom,
! [A2: nat,B3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_629_le__supE,axiom,
! [A2: set_nat,B3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_set_nat @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_630_le__supE,axiom,
! [A2: set_a,B3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
=> ~ ( ord_less_eq_set_a @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_631_le__supI,axiom,
! [A2: nat,X2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_632_le__supI,axiom,
! [A2: set_nat,X2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ( ord_less_eq_set_nat @ B3 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_633_le__supI,axiom,
! [A2: set_a,X2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ( ord_less_eq_set_a @ B3 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_634_sup__ge1,axiom,
! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_635_sup__ge1,axiom,
! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_636_sup__ge1,axiom,
! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_637_sup__ge2,axiom,
! [Y4: nat,X2: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_638_sup__ge2,axiom,
! [Y4: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_639_sup__ge2,axiom,
! [Y4: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_640_le__supI1,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_641_le__supI1,axiom,
! [X2: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_642_le__supI1,axiom,
! [X2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_643_le__supI2,axiom,
! [X2: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_644_le__supI2,axiom,
! [X2: set_nat,B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B3 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_645_le__supI2,axiom,
! [X2: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X2 @ B3 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_646_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_647_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_648_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_649_sup__mono,axiom,
! [A2: nat,C: nat,B3: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_650_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_651_sup__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_652_sup__least,axiom,
! [Y4: nat,X2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y4 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_653_sup__least,axiom,
! [Y4: set_nat,X2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y4 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_654_sup__least,axiom,
! [Y4: set_a,X2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y4 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_655_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( sup_sup_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_656_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( sup_sup_set_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_657_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y5: set_a] :
( ( sup_sup_set_a @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_658_sup_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_659_sup_OorderE,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_660_sup_OorderE,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_661_sup_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_662_sup_OorderI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B3 ) )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_663_sup_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_664_sup__unique,axiom,
! [F3: nat > nat > nat,X2: nat,Y4: nat] :
( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( F3 @ X @ Y ) )
=> ( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F3 @ X @ Y ) )
=> ( ! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( F3 @ Y @ Z ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_665_sup__unique,axiom,
! [F3: set_nat > set_nat > set_nat,X2: set_nat,Y4: set_nat] :
( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( F3 @ X @ Y ) )
=> ( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F3 @ X @ Y ) )
=> ( ! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( F3 @ Y @ Z ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_666_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X2: set_a,Y4: set_a] :
( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( F3 @ X @ Y ) )
=> ( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ Y @ ( F3 @ X @ Y ) )
=> ( ! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( F3 @ Y @ Z ) @ X ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y4 )
= ( F3 @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_667_sup_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_668_sup_Oabsorb1,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_669_sup_Oabsorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_670_sup_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_671_sup_Oabsorb2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( sup_sup_set_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_672_sup_Oabsorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_673_sup__absorb1,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_674_sup__absorb1,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_675_sup__absorb1,axiom,
! [Y4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_676_sup__absorb2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_677_sup__absorb2,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( sup_sup_set_nat @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_678_sup__absorb2,axiom,
! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( sup_sup_set_a @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_679_sup_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_680_sup_OboundedE,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_681_sup_OboundedE,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_682_sup_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_683_sup_OboundedI,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_684_sup_OboundedI,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_685_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_686_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_687_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A5: set_a] :
( A5
= ( sup_sup_set_a @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_688_sup_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_689_sup_Ocobounded1,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_690_sup_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_691_sup_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_692_sup_Ocobounded2,axiom,
! [B3: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_693_sup_Ocobounded2,axiom,
! [B3: set_a,A2: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_694_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_695_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_696_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A5: set_a] :
( ( sup_sup_set_a @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_697_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( sup_sup_nat @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_698_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_699_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B2: set_a] :
( ( sup_sup_set_a @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_700_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_701_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_702_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_703_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_704_sup_OcoboundedI2,axiom,
! [C: set_nat,B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_705_sup_OcoboundedI2,axiom,
! [C: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_706_distrib__inf__le,axiom,
! [X2: nat,Y4: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ ( inf_inf_nat @ X2 @ Z3 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y4 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_707_distrib__inf__le,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ ( inf_inf_set_nat @ X2 @ Z3 ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_708_distrib__inf__le,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y4 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_709_distrib__sup__le,axiom,
! [X2: nat,Y4: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y4 ) @ ( sup_sup_nat @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_710_distrib__sup__le,axiom,
! [X2: set_nat,Y4: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y4 @ Z3 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ ( sup_sup_set_nat @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_711_distrib__sup__le,axiom,
! [X2: set_a,Y4: set_a,Z3: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_712_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_713_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_714_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_715_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_716_one__hom_Ointro,axiom,
! [Hom: nat > nat] :
( ( ( Hom @ one_one_nat )
= one_one_nat )
=> ( ring_one_hom_nat_nat @ Hom ) ) ).
% one_hom.intro
thf(fact_717_one__hom_Ointro,axiom,
! [Hom: nat > a] :
( ( ( Hom @ one_one_nat )
= one_one_a )
=> ( ring_one_hom_nat_a @ Hom ) ) ).
% one_hom.intro
thf(fact_718_one__hom_Ointro,axiom,
! [Hom: a > nat] :
( ( ( Hom @ one_one_a )
= one_one_nat )
=> ( ring_one_hom_a_nat @ Hom ) ) ).
% one_hom.intro
thf(fact_719_one__hom_Ointro,axiom,
! [Hom: a > a] :
( ( ( Hom @ one_one_a )
= one_one_a )
=> ( ring_one_hom_a_a @ Hom ) ) ).
% one_hom.intro
thf(fact_720_one__hom_Ohom__one,axiom,
! [Hom: nat > nat] :
( ( ring_one_hom_nat_nat @ Hom )
=> ( ( Hom @ one_one_nat )
= one_one_nat ) ) ).
% one_hom.hom_one
thf(fact_721_one__hom_Ohom__one,axiom,
! [Hom: nat > a] :
( ( ring_one_hom_nat_a @ Hom )
=> ( ( Hom @ one_one_nat )
= one_one_a ) ) ).
% one_hom.hom_one
thf(fact_722_one__hom_Ohom__one,axiom,
! [Hom: a > nat] :
( ( ring_one_hom_a_nat @ Hom )
=> ( ( Hom @ one_one_a )
= one_one_nat ) ) ).
% one_hom.hom_one
thf(fact_723_one__hom_Ohom__one,axiom,
! [Hom: a > a] :
( ( ring_one_hom_a_a @ Hom )
=> ( ( Hom @ one_one_a )
= one_one_a ) ) ).
% one_hom.hom_one
thf(fact_724_one__hom__def,axiom,
( ring_one_hom_nat_nat
= ( ^ [Hom2: nat > nat] :
( ( Hom2 @ one_one_nat )
= one_one_nat ) ) ) ).
% one_hom_def
thf(fact_725_one__hom__def,axiom,
( ring_one_hom_nat_a
= ( ^ [Hom2: nat > a] :
( ( Hom2 @ one_one_nat )
= one_one_a ) ) ) ).
% one_hom_def
thf(fact_726_one__hom__def,axiom,
( ring_one_hom_a_nat
= ( ^ [Hom2: a > nat] :
( ( Hom2 @ one_one_a )
= one_one_nat ) ) ) ).
% one_hom_def
thf(fact_727_one__hom__def,axiom,
( ring_one_hom_a_a
= ( ^ [Hom2: a > a] :
( ( Hom2 @ one_one_a )
= one_one_a ) ) ) ).
% one_hom_def
thf(fact_728_zero__hom_Ointro,axiom,
! [Hom: nat > nat] :
( ( ( Hom @ zero_zero_nat )
= zero_zero_nat )
=> ( ring_z4445335182927245238at_nat @ Hom ) ) ).
% zero_hom.intro
thf(fact_729_zero__hom_Ointro,axiom,
! [Hom: nat > a] :
( ( ( Hom @ zero_zero_nat )
= zero_zero_a )
=> ( ring_zero_hom_nat_a @ Hom ) ) ).
% zero_hom.intro
thf(fact_730_zero__hom_Ointro,axiom,
! [Hom: a > nat] :
( ( ( Hom @ zero_zero_a )
= zero_zero_nat )
=> ( ring_zero_hom_a_nat @ Hom ) ) ).
% zero_hom.intro
thf(fact_731_zero__hom_Ointro,axiom,
! [Hom: a > a] :
( ( ( Hom @ zero_zero_a )
= zero_zero_a )
=> ( ring_zero_hom_a_a @ Hom ) ) ).
% zero_hom.intro
thf(fact_732_zero__hom_Ohom__zero,axiom,
! [Hom: nat > nat] :
( ( ring_z4445335182927245238at_nat @ Hom )
=> ( ( Hom @ zero_zero_nat )
= zero_zero_nat ) ) ).
% zero_hom.hom_zero
thf(fact_733_zero__hom_Ohom__zero,axiom,
! [Hom: nat > a] :
( ( ring_zero_hom_nat_a @ Hom )
=> ( ( Hom @ zero_zero_nat )
= zero_zero_a ) ) ).
% zero_hom.hom_zero
thf(fact_734_zero__hom_Ohom__zero,axiom,
! [Hom: a > nat] :
( ( ring_zero_hom_a_nat @ Hom )
=> ( ( Hom @ zero_zero_a )
= zero_zero_nat ) ) ).
% zero_hom.hom_zero
thf(fact_735_zero__hom_Ohom__zero,axiom,
! [Hom: a > a] :
( ( ring_zero_hom_a_a @ Hom )
=> ( ( Hom @ zero_zero_a )
= zero_zero_a ) ) ).
% zero_hom.hom_zero
thf(fact_736_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_737_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_738_zero__hom__def,axiom,
( ring_z4445335182927245238at_nat
= ( ^ [Hom2: nat > nat] :
( ( Hom2 @ zero_zero_nat )
= zero_zero_nat ) ) ) ).
% zero_hom_def
thf(fact_739_zero__hom__def,axiom,
( ring_zero_hom_nat_a
= ( ^ [Hom2: nat > a] :
( ( Hom2 @ zero_zero_nat )
= zero_zero_a ) ) ) ).
% zero_hom_def
thf(fact_740_zero__hom__def,axiom,
( ring_zero_hom_a_nat
= ( ^ [Hom2: a > nat] :
( ( Hom2 @ zero_zero_a )
= zero_zero_nat ) ) ) ).
% zero_hom_def
thf(fact_741_zero__hom__def,axiom,
( ring_zero_hom_a_a
= ( ^ [Hom2: a > a] :
( ( Hom2 @ zero_zero_a )
= zero_zero_a ) ) ) ).
% zero_hom_def
thf(fact_742_Int__mono2,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ) ).
% Int_mono2
thf(fact_743_Int__mono2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 ) ) ) ).
% Int_mono2
thf(fact_744_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_745_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_746_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_747_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_748_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_749_diff__self,axiom,
! [A2: a] :
( ( minus_minus_a @ A2 @ A2 )
= zero_zero_a ) ).
% diff_self
thf(fact_750_diff__0__right,axiom,
! [A2: a] :
( ( minus_minus_a @ A2 @ zero_zero_a )
= A2 ) ).
% diff_0_right
thf(fact_751_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_752_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_753_diff__zero,axiom,
! [A2: a] :
( ( minus_minus_a @ A2 @ zero_zero_a )
= A2 ) ).
% diff_zero
thf(fact_754_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_755_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: a] :
( ( minus_minus_a @ A2 @ A2 )
= zero_zero_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_756_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_757_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_758_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_759_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_760_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_761_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_762_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_763_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_764_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_765_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_766_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_767_Diff__insert0,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_768_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_769_insert__Diff1,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_770_diff__numeral__special_I9_J,axiom,
( ( minus_minus_a @ one_one_a @ one_one_a )
= zero_zero_a ) ).
% diff_numeral_special(9)
thf(fact_771_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_772_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_773_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_774_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_775_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_776_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_777_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_778_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_779_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A5: a,B2: a] :
( ( minus_minus_a @ A5 @ B2 )
= zero_zero_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_780_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_781_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_782_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_783_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_784_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_785_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_786_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_787_Diff__mono,axiom,
! [A: set_a,C2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_788_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_789_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_790_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_791_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_792_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_793_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_794_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_795_insert__Diff__if,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_796_diff__shunt__var,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_797_diff__shunt__var,axiom,
! [X2: set_a,Y4: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_798_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_799_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_800_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_801_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_802_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_803_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_804_Diff__insert__absorb,axiom,
! [X2: a,A: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_805_Diff__insert__absorb,axiom,
! [X2: nat,A: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_806_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_807_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_808_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_809_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_810_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_811_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_812_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_813_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_814_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_815_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_816_subset__insert__iff,axiom,
! [A: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B ) )
= ( ( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_817_subset__insert__iff,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_818_Diff__single__insert,axiom,
! [A: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_819_Diff__single__insert,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_820_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_821_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_822_infinite__coinduct,axiom,
! [X3: set_a > $o,A: set_a] :
( ( X3 @ A )
=> ( ! [A4: set_a] :
( ( X3 @ A4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A4 )
& ( ( X3 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_823_infinite__coinduct,axiom,
! [X3: set_nat > $o,A: set_nat] :
( ( X3 @ A )
=> ( ! [A4: set_nat] :
( ( X3 @ A4 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( X3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_824_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A3: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A3 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_825_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A3: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_826_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_827_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_828_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_829_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_830_Inf__fin_Oremove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_831_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_832_Sup__fin_Oremove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_833_set__diff__non__empty__not__subset,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
=> ( ( C2 != bot_bot_set_nat )
=> ( ( A != bot_bot_set_nat )
=> ( ( B != bot_bot_set_nat )
=> ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_834_set__diff__non__empty__not__subset,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
=> ( ( C2 != bot_bot_set_a )
=> ( ( A != bot_bot_set_a )
=> ( ( B != bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_835_remove__def,axiom,
( remove_a
= ( ^ [X4: a,A6: set_a] : ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_836_remove__def,axiom,
( remove_nat
= ( ^ [X4: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_837_member__remove,axiom,
! [X2: nat,Y4: nat,A: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y4 @ A ) )
= ( ( member_nat @ X2 @ A )
& ( X2 != Y4 ) ) ) ).
% member_remove
thf(fact_838_member__remove,axiom,
! [X2: a,Y4: a,A: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y4 @ A ) )
= ( ( member_a @ X2 @ A )
& ( X2 != Y4 ) ) ) ).
% member_remove
thf(fact_839_inj__on__Un,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( inj_on_nat_nat @ F3 @ A )
& ( inj_on_nat_nat @ F3 @ B )
& ( ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ B ) ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ B @ A ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_840_inj__on__insert,axiom,
! [F3: a > nat,A2: a,A: set_a] :
( ( inj_on_a_nat @ F3 @ ( insert_a @ A2 @ A ) )
= ( ( inj_on_a_nat @ F3 @ A )
& ~ ( member_nat @ ( F3 @ A2 ) @ ( image_a_nat @ F3 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_841_inj__on__insert,axiom,
! [F3: a > a,A2: a,A: set_a] :
( ( inj_on_a_a @ F3 @ ( insert_a @ A2 @ A ) )
= ( ( inj_on_a_a @ F3 @ A )
& ~ ( member_a @ ( F3 @ A2 ) @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_842_inj__on__insert,axiom,
! [F3: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_nat @ F3 @ A )
& ~ ( member_nat @ ( F3 @ A2 ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_843_inj__on__insert,axiom,
! [F3: nat > a,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F3 @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_a @ F3 @ A )
& ~ ( member_a @ ( F3 @ A2 ) @ ( image_nat_a @ F3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_844_image__eqI,axiom,
! [B3: nat,F3: nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F3 @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_845_image__eqI,axiom,
! [B3: a,F3: nat > a,X2: nat,A: set_nat] :
( ( B3
= ( F3 @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_a @ B3 @ ( image_nat_a @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_846_image__eqI,axiom,
! [B3: nat,F3: a > nat,X2: a,A: set_a] :
( ( B3
= ( F3 @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat @ B3 @ ( image_a_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_847_image__eqI,axiom,
! [B3: a,F3: a > a,X2: a,A: set_a] :
( ( B3
= ( F3 @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B3 @ ( image_a_a @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_848_image__is__empty,axiom,
! [F3: a > a,A: set_a] :
( ( ( image_a_a @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_849_image__is__empty,axiom,
! [F3: nat > a,A: set_nat] :
( ( ( image_nat_a @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_850_image__is__empty,axiom,
! [F3: a > nat,A: set_a] :
( ( ( image_a_nat @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_851_image__is__empty,axiom,
! [F3: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_852_empty__is__image,axiom,
! [F3: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_853_empty__is__image,axiom,
! [F3: nat > a,A: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_854_empty__is__image,axiom,
! [F3: a > nat,A: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_855_empty__is__image,axiom,
! [F3: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_856_image__empty,axiom,
! [F3: a > a] :
( ( image_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_857_image__empty,axiom,
! [F3: a > nat] :
( ( image_a_nat @ F3 @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_858_image__empty,axiom,
! [F3: nat > a] :
( ( image_nat_a @ F3 @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_859_image__empty,axiom,
! [F3: nat > nat] :
( ( image_nat_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_860_finite__imageI,axiom,
! [F: set_a,H: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_861_finite__imageI,axiom,
! [F: set_a,H: a > nat] :
( ( finite_finite_a @ F )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_862_finite__imageI,axiom,
! [F: set_nat,H: nat > a] :
( ( finite_finite_nat @ F )
=> ( finite_finite_a @ ( image_nat_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_863_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_864_insert__image,axiom,
! [X2: nat,A: set_nat,F3: nat > a] :
( ( member_nat @ X2 @ A )
=> ( ( insert_a @ ( F3 @ X2 ) @ ( image_nat_a @ F3 @ A ) )
= ( image_nat_a @ F3 @ A ) ) ) ).
% insert_image
thf(fact_865_insert__image,axiom,
! [X2: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( insert_nat @ ( F3 @ X2 ) @ ( image_nat_nat @ F3 @ A ) )
= ( image_nat_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_866_insert__image,axiom,
! [X2: a,A: set_a,F3: a > a] :
( ( member_a @ X2 @ A )
=> ( ( insert_a @ ( F3 @ X2 ) @ ( image_a_a @ F3 @ A ) )
= ( image_a_a @ F3 @ A ) ) ) ).
% insert_image
thf(fact_867_insert__image,axiom,
! [X2: a,A: set_a,F3: a > nat] :
( ( member_a @ X2 @ A )
=> ( ( insert_nat @ ( F3 @ X2 ) @ ( image_a_nat @ F3 @ A ) )
= ( image_a_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_868_image__insert,axiom,
! [F3: a > a,A2: a,B: set_a] :
( ( image_a_a @ F3 @ ( insert_a @ A2 @ B ) )
= ( insert_a @ ( F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_869_image__insert,axiom,
! [F3: a > nat,A2: a,B: set_a] :
( ( image_a_nat @ F3 @ ( insert_a @ A2 @ B ) )
= ( insert_nat @ ( F3 @ A2 ) @ ( image_a_nat @ F3 @ B ) ) ) ).
% image_insert
thf(fact_870_image__insert,axiom,
! [F3: nat > a,A2: nat,B: set_nat] :
( ( image_nat_a @ F3 @ ( insert_nat @ A2 @ B ) )
= ( insert_a @ ( F3 @ A2 ) @ ( image_nat_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_871_image__insert,axiom,
! [F3: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_insert
thf(fact_872_finite__image__iff,axiom,
! [F3: a > a,A: set_a] :
( ( inj_on_a_a @ F3 @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_873_finite__image__iff,axiom,
! [F3: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F3 @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_874_finite__image__iff,axiom,
! [F3: a > nat,A: set_a] :
( ( inj_on_a_nat @ F3 @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_875_finite__image__iff,axiom,
! [F3: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_876_map__mat__elements,axiom,
! [F3: nat > nat,A: mat_nat] :
( ( elements_mat_nat @ ( map_mat_nat_nat @ F3 @ A ) )
= ( image_nat_nat @ F3 @ ( elements_mat_nat @ A ) ) ) ).
% map_mat_elements
thf(fact_877_map__mat__elements,axiom,
! [F3: a > a,A: mat_a] :
( ( elements_mat_a @ ( map_mat_a_a @ F3 @ A ) )
= ( image_a_a @ F3 @ ( elements_mat_a @ A ) ) ) ).
% map_mat_elements
thf(fact_878_subset__image__iff,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_879_subset__image__iff,axiom,
! [B: set_nat,F3: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_880_subset__image__iff,axiom,
! [B: set_a,F3: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_881_subset__image__iff,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_882_image__subset__iff,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F3 @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_883_subset__imageE,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_884_subset__imageE,axiom,
! [B: set_nat,F3: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_885_subset__imageE,axiom,
! [B: set_a,F3: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_886_subset__imageE,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_887_image__subsetI,axiom,
! [A: set_nat,F3: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_888_image__subsetI,axiom,
! [A: set_a,F3: a > nat,B: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_889_image__subsetI,axiom,
! [A: set_nat,F3: nat > a,B: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F3 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_890_image__subsetI,axiom,
! [A: set_a,F3: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F3 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_891_image__mono,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_mono
thf(fact_892_image__mono,axiom,
! [A: set_nat,B: set_nat,F3: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ ( image_nat_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_893_image__mono,axiom,
! [A: set_a,B: set_a,F3: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ ( image_a_nat @ F3 @ B ) ) ) ).
% image_mono
thf(fact_894_image__mono,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_895_all__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F3 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_896_all__subset__image,axiom,
! [F3: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F3 @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_nat @ F3 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_897_all__subset__image,axiom,
! [F3: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F3 @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_a @ F3 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_898_all__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F3 @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_a @ F3 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_899_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat,F3: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F3 @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_900_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: a,F3: nat > a] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F3 @ X2 ) )
=> ( member_a @ B3 @ ( image_nat_a @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_901_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: nat,F3: a > nat] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F3 @ X2 ) )
=> ( member_nat @ B3 @ ( image_a_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_902_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: a,F3: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F3 @ X2 ) )
=> ( member_a @ B3 @ ( image_a_a @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_903_ball__imageD,axiom,
! [F3: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F3 @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_904_image__cong,axiom,
! [M: set_nat,N2: set_nat,F3: nat > nat,G2: nat > nat] :
( ( M = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F3 @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_nat @ F3 @ M )
= ( image_nat_nat @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_905_bex__imageD,axiom,
! [F3: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F3 @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F3 @ X ) ) ) ) ).
% bex_imageD
thf(fact_906_image__iff,axiom,
! [Z3: nat,F3: nat > nat,A: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F3 @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z3
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_907_imageI,axiom,
! [X2: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ ( image_nat_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_908_imageI,axiom,
! [X2: nat,A: set_nat,F3: nat > a] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F3 @ X2 ) @ ( image_nat_a @ F3 @ A ) ) ) ).
% imageI
thf(fact_909_imageI,axiom,
! [X2: a,A: set_a,F3: a > nat] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ ( image_a_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_910_imageI,axiom,
! [X2: a,A: set_a,F3: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F3 @ X2 ) @ ( image_a_a @ F3 @ A ) ) ) ).
% imageI
thf(fact_911_image__Un,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Un
thf(fact_912_all__finite__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_nat @ F3 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_913_all__finite__subset__image,axiom,
! [F3: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F3 @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P @ ( image_a_nat @ F3 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_914_all__finite__subset__image,axiom,
! [F3: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F3 @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_a @ F3 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_915_all__finite__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F3 @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P @ ( image_a_a @ F3 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_916_ex__finite__subset__image,axiom,
! [F3: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_nat @ F3 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_917_ex__finite__subset__image,axiom,
! [F3: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F3 @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P @ ( image_a_nat @ F3 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_918_ex__finite__subset__image,axiom,
! [F3: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F3 @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_a @ F3 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_919_ex__finite__subset__image,axiom,
! [F3: a > a,A: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F3 @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P @ ( image_a_a @ F3 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_920_finite__subset__image,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_921_finite__subset__image,axiom,
! [B: set_nat,F3: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_922_finite__subset__image,axiom,
! [B: set_a,F3: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_923_finite__subset__image,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_924_finite__surj,axiom,
! [A: set_a,B: set_nat,F3: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_925_finite__surj,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_926_finite__surj,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_927_finite__surj,axiom,
! [A: set_nat,B: set_a,F3: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_928_image__Int__subset,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_929_image__diff__subset,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_930_inj__on__image__eq__iff,axiom,
! [F3: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( ( image_nat_nat @ F3 @ A )
= ( image_nat_nat @ F3 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_931_inj__on__image__mem__iff,axiom,
! [F3: nat > nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F3 @ A2 ) @ ( image_nat_nat @ F3 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_932_inj__on__image__mem__iff,axiom,
! [F3: nat > a,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F3 @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_a @ ( F3 @ A2 ) @ ( image_nat_a @ F3 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_933_inj__on__image__mem__iff,axiom,
! [F3: a > nat,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_nat @ F3 @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_nat @ ( F3 @ A2 ) @ ( image_a_nat @ F3 @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_934_inj__on__image__mem__iff,axiom,
! [F3: a > a,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F3 @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ ( F3 @ A2 ) @ ( image_a_a @ F3 @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_935_finite__imageD,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
=> ( ( inj_on_a_a @ F3 @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_936_finite__imageD,axiom,
! [F3: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
=> ( ( inj_on_nat_a @ F3 @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_937_finite__imageD,axiom,
! [F3: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
=> ( ( inj_on_a_nat @ F3 @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_938_finite__imageD,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ( inj_on_nat_nat @ F3 @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_939_inj__img__insertE,axiom,
! [F3: nat > nat,A: set_nat,X2: nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ A )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ B )
= ( image_nat_nat @ F3 @ A ) )
=> ~ ! [X6: nat,A8: set_nat] :
( ~ ( member_nat @ X6 @ A8 )
=> ( ( A
= ( insert_nat @ X6 @ A8 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B
!= ( image_nat_nat @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_940_inj__img__insertE,axiom,
! [F3: a > nat,A: set_a,X2: nat,B: set_nat] :
( ( inj_on_a_nat @ F3 @ A )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ B )
= ( image_a_nat @ F3 @ A ) )
=> ~ ! [X6: a,A8: set_a] :
( ~ ( member_a @ X6 @ A8 )
=> ( ( A
= ( insert_a @ X6 @ A8 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B
!= ( image_a_nat @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_941_inj__img__insertE,axiom,
! [F3: nat > a,A: set_nat,X2: a,B: set_a] :
( ( inj_on_nat_a @ F3 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ B )
= ( image_nat_a @ F3 @ A ) )
=> ~ ! [X6: nat,A8: set_nat] :
( ~ ( member_nat @ X6 @ A8 )
=> ( ( A
= ( insert_nat @ X6 @ A8 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B
!= ( image_nat_a @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_942_inj__img__insertE,axiom,
! [F3: a > a,A: set_a,X2: a,B: set_a] :
( ( inj_on_a_a @ F3 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ B )
= ( image_a_a @ F3 @ A ) )
=> ~ ! [X6: a,A8: set_a] :
( ~ ( member_a @ X6 @ A8 )
=> ( ( A
= ( insert_a @ X6 @ A8 ) )
=> ( ( X2
= ( F3 @ X6 ) )
=> ( B
!= ( image_a_a @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_943_the__elem__image__unique,axiom,
! [A: set_nat,F3: nat > nat,X2: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A )
=> ( ( F3 @ Y )
= ( F3 @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F3 @ A ) )
= ( F3 @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_944_endo__inj__surj,axiom,
! [A: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ A )
=> ( ( inj_on_nat_nat @ F3 @ A )
=> ( ( image_nat_nat @ F3 @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_945_endo__inj__surj,axiom,
! [A: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ A )
=> ( ( inj_on_a_a @ F3 @ A )
=> ( ( image_a_a @ F3 @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_946_Finite__Set_Oinj__on__finite,axiom,
! [F3: a > nat,A: set_a,B: set_nat] :
( ( inj_on_a_nat @ F3 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_a @ A ) ) ) ) ).
% Finite_Set.inj_on_finite
thf(fact_947_Finite__Set_Oinj__on__finite,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ) ).
% Finite_Set.inj_on_finite
thf(fact_948_Finite__Set_Oinj__on__finite,axiom,
! [F3: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F3 @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ) ).
% Finite_Set.inj_on_finite
thf(fact_949_Finite__Set_Oinj__on__finite,axiom,
! [F3: nat > a,A: set_nat,B: set_a] :
( ( inj_on_nat_a @ F3 @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_nat @ A ) ) ) ) ).
% Finite_Set.inj_on_finite
thf(fact_950_finite__surj__inj,axiom,
! [A: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F3 @ A ) )
=> ( inj_on_nat_nat @ F3 @ A ) ) ) ).
% finite_surj_inj
thf(fact_951_finite__surj__inj,axiom,
! [A: set_a,F3: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F3 @ A ) )
=> ( inj_on_a_a @ F3 @ A ) ) ) ).
% finite_surj_inj
thf(fact_952_in__image__insert__iff,axiom,
! [B: set_set_a,X2: a,A: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X2 @ C4 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B ) )
= ( ( member_a @ X2 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_953_in__image__insert__iff,axiom,
! [B: set_set_nat,X2: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X2 @ C4 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B ) )
= ( ( member_nat @ X2 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_954_inj__on__image__Int,axiom,
! [F3: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_nat_nat @ F3 @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_955_inj__on__image__set__diff,axiom,
! [F3: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_956_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y: nat] :
( ( H @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N2 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_957_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y: nat] :
( ( H @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N2 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_958_inj__on__iff__surj,axiom,
! [A: set_a,A9: set_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F4: a > nat] :
( ( inj_on_a_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F4 @ A ) @ A9 ) ) )
= ( ? [G3: nat > a] :
( ( image_nat_a @ G3 @ A9 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_959_inj__on__iff__surj,axiom,
! [A: set_nat,A9: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ A9 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat @ G3 @ A9 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_960_inj__on__iff__surj,axiom,
! [A: set_a,A9: set_a] :
( ( A != bot_bot_set_a )
=> ( ( ? [F4: a > a] :
( ( inj_on_a_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F4 @ A ) @ A9 ) ) )
= ( ? [G3: a > a] :
( ( image_a_a @ G3 @ A9 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_961_inj__on__iff__surj,axiom,
! [A: set_nat,A9: set_a] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > a] :
( ( inj_on_nat_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ A ) @ A9 ) ) )
= ( ? [G3: a > nat] :
( ( image_a_nat @ G3 @ A9 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_962_subset__image__inj,axiom,
! [S: set_nat,F3: nat > nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F3 @ T ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T )
& ( inj_on_nat_nat @ F3 @ U )
& ( S
= ( image_nat_nat @ F3 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_963_subset__image__inj,axiom,
! [S: set_nat,F3: a > nat,T: set_a] :
( ( ord_less_eq_set_nat @ S @ ( image_a_nat @ F3 @ T ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T )
& ( inj_on_a_nat @ F3 @ U )
& ( S
= ( image_a_nat @ F3 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_964_subset__image__inj,axiom,
! [S: set_a,F3: nat > a,T: set_nat] :
( ( ord_less_eq_set_a @ S @ ( image_nat_a @ F3 @ T ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T )
& ( inj_on_nat_a @ F3 @ U )
& ( S
= ( image_nat_a @ F3 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_965_subset__image__inj,axiom,
! [S: set_a,F3: a > a,T: set_a] :
( ( ord_less_eq_set_a @ S @ ( image_a_a @ F3 @ T ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T )
& ( inj_on_a_a @ F3 @ U )
& ( S
= ( image_a_a @ F3 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_966_subset__imageE__inj,axiom,
! [B: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( ( B
= ( image_nat_nat @ F3 @ C4 ) )
=> ~ ( inj_on_nat_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE_inj
thf(fact_967_subset__imageE__inj,axiom,
! [B: set_nat,F3: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( ( B
= ( image_a_nat @ F3 @ C4 ) )
=> ~ ( inj_on_a_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE_inj
thf(fact_968_subset__imageE__inj,axiom,
! [B: set_a,F3: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( ( B
= ( image_nat_a @ F3 @ C4 ) )
=> ~ ( inj_on_nat_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE_inj
thf(fact_969_subset__imageE__inj,axiom,
! [B: set_a,F3: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( ( B
= ( image_a_a @ F3 @ C4 ) )
=> ~ ( inj_on_a_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE_inj
thf(fact_970_image__Fpow__mono,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F3 ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_971_the__inv__into__into,axiom,
! [F3: nat > nat,A: set_nat,X2: nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ A )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F3 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F3 @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_972_the__inv__into__into,axiom,
! [F3: nat > a,A: set_nat,X2: a,B: set_nat] :
( ( inj_on_nat_a @ F3 @ A )
=> ( ( member_a @ X2 @ ( image_nat_a @ F3 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_a @ A @ F3 @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_973_the__inv__into__into,axiom,
! [F3: a > nat,A: set_a,X2: nat,B: set_a] :
( ( inj_on_a_nat @ F3 @ A )
=> ( ( member_nat @ X2 @ ( image_a_nat @ F3 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_nat @ A @ F3 @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_974_the__inv__into__into,axiom,
! [F3: a > a,A: set_a,X2: a,B: set_a] :
( ( inj_on_a_a @ F3 @ A )
=> ( ( member_a @ X2 @ ( image_a_a @ F3 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_a @ A @ F3 @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_975_inf__img__fin__domE_H,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ~ ! [Y: a] :
( ( member_a @ Y @ ( image_a_a @ F3 @ A ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_976_inf__img__fin__domE_H,axiom,
! [F3: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y: a] :
( ( member_a @ Y @ ( image_nat_a @ F3 @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ F3 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_977_inf__img__fin__domE_H,axiom,
! [F3: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_a_nat @ F3 @ A ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_nat @ F3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_978_inf__img__fin__domE_H,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_nat_nat @ F3 @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_979_inf__img__fin__dom_H,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ? [X: a] :
( ( member_a @ X @ ( image_a_a @ F3 @ A ) )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_980_inf__img__fin__dom_H,axiom,
! [F3: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X: a] :
( ( member_a @ X @ ( image_nat_a @ F3 @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ F3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_981_inf__img__fin__dom_H,axiom,
! [F3: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F3 @ A ) )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_nat @ F3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_982_inf__img__fin__dom_H,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F3 @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_983_vimage__eq,axiom,
! [A2: nat,F3: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ B ) )
= ( member_nat @ ( F3 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_984_vimage__eq,axiom,
! [A2: nat,F3: nat > a,B: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ B ) )
= ( member_a @ ( F3 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_985_vimage__eq,axiom,
! [A2: a,F3: a > nat,B: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F3 @ B ) )
= ( member_nat @ ( F3 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_986_vimage__eq,axiom,
! [A2: a,F3: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F3 @ B ) )
= ( member_a @ ( F3 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_987_vimageI,axiom,
! [F3: nat > nat,A2: nat,B3: nat,B: set_nat] :
( ( ( F3 @ A2 )
= B3 )
=> ( ( member_nat @ B3 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ B ) ) ) ) ).
% vimageI
thf(fact_988_vimageI,axiom,
! [F3: a > nat,A2: a,B3: nat,B: set_nat] :
( ( ( F3 @ A2 )
= B3 )
=> ( ( member_nat @ B3 @ B )
=> ( member_a @ A2 @ ( vimage_a_nat @ F3 @ B ) ) ) ) ).
% vimageI
thf(fact_989_vimageI,axiom,
! [F3: nat > a,A2: nat,B3: a,B: set_a] :
( ( ( F3 @ A2 )
= B3 )
=> ( ( member_a @ B3 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ B ) ) ) ) ).
% vimageI
thf(fact_990_vimageI,axiom,
! [F3: a > a,A2: a,B3: a,B: set_a] :
( ( ( F3 @ A2 )
= B3 )
=> ( ( member_a @ B3 @ B )
=> ( member_a @ A2 @ ( vimage_a_a @ F3 @ B ) ) ) ) ).
% vimageI
thf(fact_991_vimage__empty,axiom,
! [F3: a > a] :
( ( vimage_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_992_vimage__empty,axiom,
! [F3: nat > a] :
( ( vimage_nat_a @ F3 @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_993_vimage__empty,axiom,
! [F3: a > nat] :
( ( vimage_a_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_994_vimage__empty,axiom,
! [F3: nat > nat] :
( ( vimage_nat_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_995_subset__vimage__iff,axiom,
! [A: set_nat,F3: nat > nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F3 @ B ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F3 @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_996_subset__vimage__iff,axiom,
! [A: set_nat,F3: nat > a,B: set_a] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_a @ F3 @ B ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_a @ ( F3 @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_997_subset__vimage__iff,axiom,
! [A: set_a,F3: a > nat,B: set_nat] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_nat @ F3 @ B ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_nat @ ( F3 @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_998_subset__vimage__iff,axiom,
! [A: set_a,F3: a > a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F3 @ B ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F3 @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_999_vimage__mono,axiom,
! [A: set_nat,B: set_nat,F3: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F3 @ A ) @ ( vimage_nat_nat @ F3 @ B ) ) ) ).
% vimage_mono
thf(fact_1000_vimage__mono,axiom,
! [A: set_nat,B: set_nat,F3: a > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_nat @ F3 @ A ) @ ( vimage_a_nat @ F3 @ B ) ) ) ).
% vimage_mono
thf(fact_1001_vimage__mono,axiom,
! [A: set_a,B: set_a,F3: nat > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_a @ F3 @ A ) @ ( vimage_nat_a @ F3 @ B ) ) ) ).
% vimage_mono
thf(fact_1002_vimage__mono,axiom,
! [A: set_a,B: set_a,F3: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F3 @ A ) @ ( vimage_a_a @ F3 @ B ) ) ) ).
% vimage_mono
thf(fact_1003_vimageI2,axiom,
! [F3: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F3 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ A ) ) ) ).
% vimageI2
thf(fact_1004_vimageI2,axiom,
! [F3: a > nat,A2: a,A: set_nat] :
( ( member_nat @ ( F3 @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_nat @ F3 @ A ) ) ) ).
% vimageI2
thf(fact_1005_vimageI2,axiom,
! [F3: nat > a,A2: nat,A: set_a] :
( ( member_a @ ( F3 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ A ) ) ) ).
% vimageI2
thf(fact_1006_vimageI2,axiom,
! [F3: a > a,A2: a,A: set_a] :
( ( member_a @ ( F3 @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_a @ F3 @ A ) ) ) ).
% vimageI2
thf(fact_1007_vimageE,axiom,
! [A2: nat,F3: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ B ) )
=> ( member_nat @ ( F3 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1008_vimageE,axiom,
! [A2: nat,F3: nat > a,B: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ B ) )
=> ( member_a @ ( F3 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1009_vimageE,axiom,
! [A2: a,F3: a > nat,B: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F3 @ B ) )
=> ( member_nat @ ( F3 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1010_vimageE,axiom,
! [A2: a,F3: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F3 @ B ) )
=> ( member_a @ ( F3 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1011_vimageD,axiom,
! [A2: nat,F3: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ A ) )
=> ( member_nat @ ( F3 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1012_vimageD,axiom,
! [A2: nat,F3: nat > a,A: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ A ) )
=> ( member_a @ ( F3 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1013_vimageD,axiom,
! [A2: a,F3: a > nat,A: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F3 @ A ) )
=> ( member_nat @ ( F3 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1014_vimageD,axiom,
! [A2: a,F3: a > a,A: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F3 @ A ) )
=> ( member_a @ ( F3 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1015_image__subset__iff__subset__vimage,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F3 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1016_image__subset__iff__subset__vimage,axiom,
! [F3: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_nat @ F3 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1017_image__subset__iff__subset__vimage,axiom,
! [F3: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_a @ F3 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1018_image__subset__iff__subset__vimage,axiom,
! [F3: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F3 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1019_image__vimage__subset,axiom,
! [F3: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ ( vimage_nat_nat @ F3 @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1020_vimage__singleton__eq,axiom,
! [A2: nat,F3: nat > a,B3: a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( ( F3 @ A2 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1021_vimage__singleton__eq,axiom,
! [A2: a,F3: a > a,B3: a] :
( ( member_a @ A2 @ ( vimage_a_a @ F3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( ( F3 @ A2 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1022_vimage__singleton__eq,axiom,
! [A2: nat,F3: nat > nat,B3: nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
= ( ( F3 @ A2 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1023_vimage__singleton__eq,axiom,
! [A2: a,F3: a > nat,B3: nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
= ( ( F3 @ A2 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1024_empty__in__Fpow,axiom,
! [A: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A ) ) ).
% empty_in_Fpow
thf(fact_1025_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_1026_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_1027_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1028_finite__vimage__IntI,axiom,
! [F: set_a,H: a > a,A: set_a] :
( ( finite_finite_a @ F )
=> ( ( inj_on_a_a @ H @ A )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H @ F ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1029_finite__vimage__IntI,axiom,
! [F: set_a,H: nat > a,A: set_nat] :
( ( finite_finite_a @ F )
=> ( ( inj_on_nat_a @ H @ A )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ H @ F ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1030_finite__vimage__IntI,axiom,
! [F: set_nat,H: a > nat,A: set_a] :
( ( finite_finite_nat @ F )
=> ( ( inj_on_a_nat @ H @ A )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_nat @ H @ F ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1031_finite__vimage__IntI,axiom,
! [F: set_nat,H: nat > nat,A: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( inj_on_nat_nat @ H @ A )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1032_inf__img__fin__domE,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ~ ! [Y: a] :
( ( member_a @ Y @ ( image_a_a @ F3 @ A ) )
=> ( finite_finite_a @ ( vimage_a_a @ F3 @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1033_inf__img__fin__domE,axiom,
! [F3: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y: a] :
( ( member_a @ Y @ ( image_nat_a @ F3 @ A ) )
=> ( finite_finite_nat @ ( vimage_nat_a @ F3 @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1034_inf__img__fin__domE,axiom,
! [F3: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_a_nat @ F3 @ A ) )
=> ( finite_finite_a @ ( vimage_a_nat @ F3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1035_inf__img__fin__domE,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_nat_nat @ F3 @ A ) )
=> ( finite_finite_nat @ ( vimage_nat_nat @ F3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1036_inf__img__fin__dom,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ? [X: a] :
( ( member_a @ X @ ( image_a_a @ F3 @ A ) )
& ~ ( finite_finite_a @ ( vimage_a_a @ F3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1037_inf__img__fin__dom,axiom,
! [F3: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X: a] :
( ( member_a @ X @ ( image_nat_a @ F3 @ A ) )
& ~ ( finite_finite_nat @ ( vimage_nat_a @ F3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1038_inf__img__fin__dom,axiom,
! [F3: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_a @ A )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F3 @ A ) )
& ~ ( finite_finite_a @ ( vimage_a_nat @ F3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1039_inf__img__fin__dom,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F3 @ A ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1040_finite__finite__vimage__IntI,axiom,
! [F: set_a,H: a > a,A: set_a] :
( ( finite_finite_a @ F )
=> ( ! [Y: a] :
( ( member_a @ Y @ F )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H @ F ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1041_finite__finite__vimage__IntI,axiom,
! [F: set_a,H: nat > a,A: set_nat] :
( ( finite_finite_a @ F )
=> ( ! [Y: a] :
( ( member_a @ Y @ F )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ H @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_a @ H @ F ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1042_finite__finite__vimage__IntI,axiom,
! [F: set_nat,H: a > nat,A: set_a] :
( ( finite_finite_nat @ F )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ F )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_nat @ H @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_nat @ H @ F ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1043_finite__finite__vimage__IntI,axiom,
! [F: set_nat,H: nat > nat,A: set_nat] :
( ( finite_finite_nat @ F )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ F )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1044_inj__on__image__Fpow,axiom,
! [F3: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F3 ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_1045_image__Pow__mono,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F3 ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_1046_vimage__subsetI,axiom,
! [F3: product_unit > nat,B: set_nat,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ B @ ( image_875570014554754200it_nat @ F3 @ A ) )
=> ( ord_le3507040750410214029t_unit @ ( vimage6253328473476588386it_nat @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1047_vimage__subsetI,axiom,
! [F3: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1048_vimage__subsetI,axiom,
! [F3: a > nat,B: set_nat,A: set_a] :
( ( inj_on_a_nat @ F3 @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) )
=> ( ord_less_eq_set_a @ ( vimage_a_nat @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1049_vimage__subsetI,axiom,
! [F3: product_unit > a,B: set_a,A: set_Product_unit] :
( ( inj_on8151663806560157602unit_a @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_a @ B @ ( image_Product_unit_a @ F3 @ A ) )
=> ( ord_le3507040750410214029t_unit @ ( vimage4490873842868098028unit_a @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1050_vimage__subsetI,axiom,
! [F3: nat > a,B: set_a,A: set_nat] :
( ( inj_on_nat_a @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_a @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1051_vimage__subsetI,axiom,
! [F3: a > a,B: set_a,A: set_a] :
( ( inj_on_a_a @ F3 @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F3 @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1052_finite__vimageD_H,axiom,
! [F3: a > nat,A: set_nat] :
( ( finite_finite_a @ ( vimage_a_nat @ F3 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_a_nat @ F3 @ top_top_set_a ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_1053_finite__vimageD_H,axiom,
! [F3: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F3 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F3 @ top_top_set_nat ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_1054_finite__vimageD_H,axiom,
! [F3: product_unit > nat,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ F3 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_1055_finite__vimageD_H,axiom,
! [F3: a > a,A: set_a] :
( ( finite_finite_a @ ( vimage_a_a @ F3 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F3 @ top_top_set_a ) )
=> ( finite_finite_a @ A ) ) ) ).
% finite_vimageD'
thf(fact_1056_finite__vimageD_H,axiom,
! [F3: nat > a,A: set_a] :
( ( finite_finite_nat @ ( vimage_nat_a @ F3 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F3 @ top_top_set_nat ) )
=> ( finite_finite_a @ A ) ) ) ).
% finite_vimageD'
thf(fact_1057_finite__vimageD_H,axiom,
! [F3: product_unit > a,A: set_a] :
( ( finite4290736615968046902t_unit @ ( vimage4490873842868098028unit_a @ F3 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_a @ A ) ) ) ).
% finite_vimageD'
thf(fact_1058_image__vimage__eq,axiom,
! [F3: nat > nat,A: set_nat] :
( ( image_nat_nat @ F3 @ ( vimage_nat_nat @ F3 @ A ) )
= ( inf_inf_set_nat @ A @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1059_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1060_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1061_UNIV__I,axiom,
! [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_1062_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1063_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1064_finite__Plus__UNIV__iff,axiom,
( ( finite2069262655233506379t_unit @ top_to1755696212014396186t_unit )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1065_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1066_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_1067_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1068_finite__Plus__UNIV__iff,axiom,
( ( finite1276461556078370925unit_a @ top_to5559247480540603964unit_a )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1069_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1070_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1071_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1072_Int__UNIV,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_1073_vimage__UNIV,axiom,
! [F3: nat > nat] :
( ( vimage_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1074_vimage__UNIV,axiom,
! [F3: product_unit > nat] :
( ( vimage6253328473476588386it_nat @ F3 @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_1075_vimage__UNIV,axiom,
! [F3: nat > product_unit] :
( ( vimage4884490618288580032t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1076_vimage__UNIV,axiom,
! [F3: product_unit > product_unit] :
( ( vimage7995052115951654139t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_1077_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1078_Pow__UNIV,axiom,
( ( pow_Product_unit @ top_to1996260823553986621t_unit )
= top_to1767297665138865437t_unit ) ).
% Pow_UNIV
thf(fact_1079_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_1080_Pow__iff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_1081_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_1082_PowI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_1083_finite__Pow__iff,axiom,
! [A: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_Pow_iff
thf(fact_1084_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_1085_Diff__UNIV,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1086_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1087_Diff__UNIV,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_1088_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_1089_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_1090_Pow__singleton__iff,axiom,
! [X3: set_a,Y6: set_a] :
( ( ( pow_a @ X3 )
= ( insert_set_a @ Y6 @ bot_bot_set_set_a ) )
= ( ( X3 = bot_bot_set_a )
& ( Y6 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1091_Pow__singleton__iff,axiom,
! [X3: set_nat,Y6: set_nat] :
( ( ( pow_nat @ X3 )
= ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y6 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_1092_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_1093_PowD,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% PowD
thf(fact_1094_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_1095_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1096_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1097_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_1098_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1099_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1100_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_1101_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1102_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1103_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_1104_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_1105_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_1106_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_1107_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_1108_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_1109_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1110_finite__prod,axiom,
( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1111_finite__prod,axiom,
( ( finite1408885517383445215t_unit @ top_to6636102223169616742t_unit )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_1112_finite__prod,axiom,
( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1113_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_1114_finite__prod,axiom,
( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_1115_finite__prod,axiom,
( ( finite616084418228309761unit_a @ top_to1216281454841048712unit_a )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1116_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1117_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_1118_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1119_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1120_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite1408885517383445215t_unit @ top_to6636102223169616742t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_1121_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1122_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_1123_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_1124_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite616084418228309761unit_a @ top_to1216281454841048712unit_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1125_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1126_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_1127_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X: a] : ( member_a @ X @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1128_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_1129_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X: product_unit] : ( member_Product_unit @ X @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_1130_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1131_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1132_UNIV__witness,axiom,
? [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_1133_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1134_Un__UNIV__left,axiom,
! [B: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_1135_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1136_Un__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_1137_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1138_Int__UNIV__left,axiom,
! [B: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1139_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_1140_Int__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% Int_UNIV_right
thf(fact_1141_insert__UNIV,axiom,
! [X2: a] :
( ( insert_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_1142_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_1143_insert__UNIV,axiom,
! [X2: product_unit] :
( ( insert_Product_unit @ X2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_1144_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1145_ex__new__if__finite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A )
=> ? [A3: a] :
~ ( member_a @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1146_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A3: nat] :
~ ( member_nat @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1147_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A3: product_unit] :
~ ( member_Product_unit @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1148_finite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_UNIV
thf(fact_1149_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1150_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1151_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_1152_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_1153_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_1154_rangeI,axiom,
! [F3: nat > nat,X2: nat] : ( member_nat @ ( F3 @ X2 ) @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1155_rangeI,axiom,
! [F3: nat > a,X2: nat] : ( member_a @ ( F3 @ X2 ) @ ( image_nat_a @ F3 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1156_rangeI,axiom,
! [F3: product_unit > nat,X2: product_unit] : ( member_nat @ ( F3 @ X2 ) @ ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_1157_rangeI,axiom,
! [F3: product_unit > a,X2: product_unit] : ( member_a @ ( F3 @ X2 ) @ ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_1158_range__eqI,axiom,
! [B3: nat,F3: nat > nat,X2: nat] :
( ( B3
= ( F3 @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1159_range__eqI,axiom,
! [B3: a,F3: nat > a,X2: nat] :
( ( B3
= ( F3 @ X2 ) )
=> ( member_a @ B3 @ ( image_nat_a @ F3 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1160_range__eqI,axiom,
! [B3: nat,F3: product_unit > nat,X2: product_unit] :
( ( B3
= ( F3 @ X2 ) )
=> ( member_nat @ B3 @ ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_1161_range__eqI,axiom,
! [B3: a,F3: product_unit > a,X2: product_unit] :
( ( B3
= ( F3 @ X2 ) )
=> ( member_a @ B3 @ ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_1162_range__inj__infinite,axiom,
! [F3: nat > a] :
( ( inj_on_nat_a @ F3 @ top_top_set_nat )
=> ~ ( finite_finite_a @ ( image_nat_a @ F3 @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_1163_range__inj__infinite,axiom,
! [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ~ ( finite_finite_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_1164_range__subsetD,axiom,
! [F3: nat > nat,B: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F3 @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1165_range__subsetD,axiom,
! [F3: product_unit > nat,B: set_nat,I: product_unit] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit ) @ B )
=> ( member_nat @ ( F3 @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1166_range__subsetD,axiom,
! [F3: nat > a,B: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ top_top_set_nat ) @ B )
=> ( member_a @ ( F3 @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1167_range__subsetD,axiom,
! [F3: product_unit > a,B: set_a,I: product_unit] :
( ( ord_less_eq_set_a @ ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit ) @ B )
=> ( member_a @ ( F3 @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1168_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_1169_infinite__countable__subset,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ? [F5: nat > a] :
( ( inj_on_nat_a @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_1170_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1171_infinite__iff__countable__subset,axiom,
! [S: set_a] :
( ( ~ ( finite_finite_a @ S ) )
= ( ? [F4: nat > a] :
( ( inj_on_nat_a @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1172_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_1173_Pow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_1174_image__Pow__surj,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F3 @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F3 ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_1175_Pow__insert,axiom,
! [A2: a,A: set_a] :
( ( pow_a @ ( insert_a @ A2 @ A ) )
= ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( image_set_a_set_a @ ( insert_a @ A2 ) @ ( pow_a @ A ) ) ) ) ).
% Pow_insert
thf(fact_1176_Pow__insert,axiom,
! [A2: nat,A: set_nat] :
( ( pow_nat @ ( insert_nat @ A2 @ A ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A2 ) @ ( pow_nat @ A ) ) ) ) ).
% Pow_insert
thf(fact_1177_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_a,X2: set_a,Y4: set_a] :
( ( ( inf_inf_set_a @ A2 @ X2 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ X2 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A2 @ Y4 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ Y4 )
= top_top_set_a )
=> ( X2 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1178_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X2: set_nat,Y4: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X2 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X2 )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y4 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y4 )
= top_top_set_nat )
=> ( X2 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1179_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Product_unit,X2: set_Product_unit,Y4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ X2 )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ X2 )
= top_to1996260823553986621t_unit )
=> ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y4 )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ Y4 )
= top_to1996260823553986621t_unit )
=> ( X2 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1180_range__eq__singletonD,axiom,
! [F3: nat > a,A2: a,X2: nat] :
( ( ( image_nat_a @ F3 @ top_top_set_nat )
= ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( ( F3 @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1181_range__eq__singletonD,axiom,
! [F3: nat > nat,A2: nat,X2: nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F3 @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1182_range__eq__singletonD,axiom,
! [F3: product_unit > a,A2: a,X2: product_unit] :
( ( ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit )
= ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( ( F3 @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1183_range__eq__singletonD,axiom,
! [F3: product_unit > nat,A2: nat,X2: product_unit] :
( ( ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F3 @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1184_inj__image__subset__iff,axiom,
! [F3: product_unit > nat,A: set_Product_unit,B: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F3 @ A ) @ ( image_875570014554754200it_nat @ F3 @ B ) )
= ( ord_le3507040750410214029t_unit @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1185_inj__image__subset__iff,axiom,
! [F3: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1186_inj__image__subset__iff,axiom,
! [F3: a > nat,A: set_a,B: set_a] :
( ( inj_on_a_nat @ F3 @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ ( image_a_nat @ F3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1187_inj__image__subset__iff,axiom,
! [F3: product_unit > a,A: set_Product_unit,B: set_Product_unit] :
( ( inj_on8151663806560157602unit_a @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_a @ ( image_Product_unit_a @ F3 @ A ) @ ( image_Product_unit_a @ F3 @ B ) )
= ( ord_le3507040750410214029t_unit @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1188_inj__image__subset__iff,axiom,
! [F3: nat > a,A: set_nat,B: set_nat] :
( ( inj_on_nat_a @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ ( image_nat_a @ F3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1189_inj__image__subset__iff,axiom,
! [F3: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F3 @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ ( image_a_a @ F3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1190_finite__UNIV__surj__inj,axiom,
! [F3: a > a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( ( image_a_a @ F3 @ top_top_set_a )
= top_top_set_a )
=> ( inj_on_a_a @ F3 @ top_top_set_a ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1191_finite__UNIV__surj__inj,axiom,
! [F3: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F3 @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1192_finite__UNIV__surj__inj,axiom,
! [F3: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( image_405062704495631173t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( inj_on8151373323710067377t_unit @ F3 @ top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1193_finite__UNIV__inj__surj,axiom,
! [F3: a > a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( inj_on_a_a @ F3 @ top_top_set_a )
=> ( ( image_a_a @ F3 @ top_top_set_a )
= top_top_set_a ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1194_finite__UNIV__inj__surj,axiom,
! [F3: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1195_finite__UNIV__inj__surj,axiom,
! [F3: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( inj_on8151373323710067377t_unit @ F3 @ top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1196_surj__vimage__empty,axiom,
! [F3: a > a,A: set_a] :
( ( ( image_a_a @ F3 @ top_top_set_a )
= top_top_set_a )
=> ( ( ( vimage_a_a @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_1197_surj__vimage__empty,axiom,
! [F3: a > nat,A: set_nat] :
( ( ( image_a_nat @ F3 @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( vimage_a_nat @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1198_surj__vimage__empty,axiom,
! [F3: a > product_unit,A: set_Product_unit] :
( ( ( image_a_Product_unit @ F3 @ top_top_set_a )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage3195509354877090634t_unit @ F3 @ A )
= bot_bot_set_a )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_1199_surj__vimage__empty,axiom,
! [F3: nat > a,A: set_a] :
( ( ( image_nat_a @ F3 @ top_top_set_nat )
= top_top_set_a )
=> ( ( ( vimage_nat_a @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_1200_surj__vimage__empty,axiom,
! [F3: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( vimage_nat_nat @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1201_surj__vimage__empty,axiom,
! [F3: nat > product_unit,A: set_Product_unit] :
( ( ( image_8730104196221521654t_unit @ F3 @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage4884490618288580032t_unit @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_1202_surj__vimage__empty,axiom,
! [F3: product_unit > a,A: set_a] :
( ( ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ( ( ( vimage4490873842868098028unit_a @ F3 @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_1203_surj__vimage__empty,axiom,
! [F3: product_unit > nat,A: set_nat] :
( ( ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( vimage6253328473476588386it_nat @ F3 @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1204_surj__vimage__empty,axiom,
! [F3: product_unit > product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage7995052115951654139t_unit @ F3 @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_1205_vimage__subsetD,axiom,
! [F3: product_unit > product_unit,B: set_Product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage7995052115951654139t_unit @ F3 @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_405062704495631173t_unit @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1206_vimage__subsetD,axiom,
! [F3: product_unit > nat,B: set_nat,A: set_Product_unit] :
( ( ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage6253328473476588386it_nat @ F3 @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_875570014554754200it_nat @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1207_vimage__subsetD,axiom,
! [F3: product_unit > a,B: set_a,A: set_Product_unit] :
( ( ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage4490873842868098028unit_a @ F3 @ B ) @ A )
=> ( ord_less_eq_set_a @ B @ ( image_Product_unit_a @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1208_vimage__subsetD,axiom,
! [F3: nat > product_unit,B: set_Product_unit,A: set_nat] :
( ( ( image_8730104196221521654t_unit @ F3 @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ ( vimage4884490618288580032t_unit @ F3 @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_8730104196221521654t_unit @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1209_vimage__subsetD,axiom,
! [F3: nat > nat,B: set_nat,A: set_nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F3 @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1210_vimage__subsetD,axiom,
! [F3: nat > a,B: set_a,A: set_nat] :
( ( ( image_nat_a @ F3 @ top_top_set_nat )
= top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_a @ F3 @ B ) @ A )
=> ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1211_vimage__subsetD,axiom,
! [F3: a > product_unit,B: set_Product_unit,A: set_a] :
( ( ( image_a_Product_unit @ F3 @ top_top_set_a )
= top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_a @ ( vimage3195509354877090634t_unit @ F3 @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_a_Product_unit @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1212_vimage__subsetD,axiom,
! [F3: a > nat,B: set_nat,A: set_a] :
( ( ( image_a_nat @ F3 @ top_top_set_a )
= top_top_set_nat )
=> ( ( ord_less_eq_set_a @ ( vimage_a_nat @ F3 @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1213_vimage__subsetD,axiom,
! [F3: a > a,B: set_a,A: set_a] :
( ( ( image_a_a @ F3 @ top_top_set_a )
= top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_a @ F3 @ B ) @ A )
=> ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1214_finite__vimageD,axiom,
! [H: a > a,F: set_a] :
( ( finite_finite_a @ ( vimage_a_a @ H @ F ) )
=> ( ( ( image_a_a @ H @ top_top_set_a )
= top_top_set_a )
=> ( finite_finite_a @ F ) ) ) ).
% finite_vimageD
thf(fact_1215_finite__vimageD,axiom,
! [H: a > nat,F: set_nat] :
( ( finite_finite_a @ ( vimage_a_nat @ H @ F ) )
=> ( ( ( image_a_nat @ H @ top_top_set_a )
= top_top_set_nat )
=> ( finite_finite_nat @ F ) ) ) ).
% finite_vimageD
thf(fact_1216_finite__vimageD,axiom,
! [H: a > product_unit,F: set_Product_unit] :
( ( finite_finite_a @ ( vimage3195509354877090634t_unit @ H @ F ) )
=> ( ( ( image_a_Product_unit @ H @ top_top_set_a )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F ) ) ) ).
% finite_vimageD
thf(fact_1217_finite__vimageD,axiom,
! [H: nat > a,F: set_a] :
( ( finite_finite_nat @ ( vimage_nat_a @ H @ F ) )
=> ( ( ( image_nat_a @ H @ top_top_set_nat )
= top_top_set_a )
=> ( finite_finite_a @ F ) ) ) ).
% finite_vimageD
thf(fact_1218_finite__vimageD,axiom,
! [H: nat > nat,F: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F ) )
=> ( ( ( image_nat_nat @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F ) ) ) ).
% finite_vimageD
thf(fact_1219_finite__vimageD,axiom,
! [H: nat > product_unit,F: set_Product_unit] :
( ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ H @ F ) )
=> ( ( ( image_8730104196221521654t_unit @ H @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F ) ) ) ).
% finite_vimageD
thf(fact_1220_finite__vimageD,axiom,
! [H: product_unit > a,F: set_a] :
( ( finite4290736615968046902t_unit @ ( vimage4490873842868098028unit_a @ H @ F ) )
=> ( ( ( image_Product_unit_a @ H @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ( finite_finite_a @ F ) ) ) ).
% finite_vimageD
thf(fact_1221_finite__vimageD,axiom,
! [H: product_unit > nat,F: set_nat] :
( ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ H @ F ) )
=> ( ( ( image_875570014554754200it_nat @ H @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( finite_finite_nat @ F ) ) ) ).
% finite_vimageD
thf(fact_1222_finite__vimageD,axiom,
! [H: product_unit > product_unit,F: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( vimage7995052115951654139t_unit @ H @ F ) )
=> ( ( ( image_405062704495631173t_unit @ H @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F ) ) ) ).
% finite_vimageD
thf(fact_1223_finite__vimageI,axiom,
! [F: set_a,H: a > a] :
( ( finite_finite_a @ F )
=> ( ( inj_on_a_a @ H @ top_top_set_a )
=> ( finite_finite_a @ ( vimage_a_a @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1224_finite__vimageI,axiom,
! [F: set_nat,H: a > nat] :
( ( finite_finite_nat @ F )
=> ( ( inj_on_a_nat @ H @ top_top_set_a )
=> ( finite_finite_a @ ( vimage_a_nat @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1225_finite__vimageI,axiom,
! [F: set_a,H: nat > a] :
( ( finite_finite_a @ F )
=> ( ( inj_on_nat_a @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage_nat_a @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1226_finite__vimageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( ( inj_on_nat_nat @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage_nat_nat @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1227_finite__vimageI,axiom,
! [F: set_a,H: product_unit > a] :
( ( finite_finite_a @ F )
=> ( ( inj_on8151663806560157602unit_a @ H @ top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ ( vimage4490873842868098028unit_a @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1228_finite__vimageI,axiom,
! [F: set_nat,H: product_unit > nat] :
( ( finite_finite_nat @ F )
=> ( ( inj_on8430439091780834860it_nat @ H @ top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ H @ F ) ) ) ) ).
% finite_vimageI
thf(fact_1229_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_1230_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1231_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_1232_card__vimage__inj,axiom,
! [F3: nat > product_unit,A: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F3 @ top_top_set_nat )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_8730104196221521654t_unit @ F3 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage4884490618288580032t_unit @ F3 @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1233_card__vimage__inj,axiom,
! [F3: product_unit > product_unit,A: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_405062704495631173t_unit @ F3 @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage7995052115951654139t_unit @ F3 @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1234_card__vimage__inj,axiom,
! [F3: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F3 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F3 @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1235_card__vimage__inj,axiom,
! [F3: product_unit > nat,A: set_nat] :
( ( inj_on8430439091780834860it_nat @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ A @ ( image_875570014554754200it_nat @ F3 @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage6253328473476588386it_nat @ F3 @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1236_card__vimage__inj,axiom,
! [F3: nat > a,A: set_a] :
( ( inj_on_nat_a @ F3 @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F3 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_a @ F3 @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1237_card__vimage__inj,axiom,
! [F3: product_unit > a,A: set_a] :
( ( inj_on8151663806560157602unit_a @ F3 @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_a @ A @ ( image_Product_unit_a @ F3 @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage4490873842868098028unit_a @ F3 @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1238_Sup__fin_Oeq__fold,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X2 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1239_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_1240_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1241_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1242_card_Oinfinite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1243_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1244_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1245_card__0__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_1246_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_1247_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1248_card__Diff__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ~ ( member_a @ A2 @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1249_card__Diff__insert,axiom,
! [A2: product_unit,A: set_Product_unit,B: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ A2 @ B )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1250_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1251_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_1252_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1253_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_1254_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1255_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1256_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1257_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_1258_is__singleton__altdef,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A6: set_Product_unit] :
( ( finite410649719033368117t_unit @ A6 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1259_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( ( finite_card_nat @ A6 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1260_fold__closed__eq,axiom,
! [A: set_a,B: set_a,F3: a > a > a,G2: a > a > a,Z3: a] :
( ! [A3: a,B6: a] :
( ( member_a @ A3 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( F3 @ A3 @ B6 )
= ( G2 @ A3 @ B6 ) ) ) )
=> ( ! [A3: a,B6: a] :
( ( member_a @ A3 @ A )
=> ( ( member_a @ B6 @ B )
=> ( member_a @ ( G2 @ A3 @ B6 ) @ B ) ) )
=> ( ( member_a @ Z3 @ B )
=> ( ( finite_fold_a_a @ F3 @ Z3 @ A )
= ( finite_fold_a_a @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1261_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1262_inf__pigeonhole__principle,axiom,
! [N: nat,F3: nat > nat > $o] :
( ! [K: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F3 @ K @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K2: nat] :
? [K3: nat] :
( ( ord_less_eq_nat @ K2 @ K3 )
& ( F3 @ K3 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1263_unbounded__k__infinite,axiom,
! [K4: nat,S: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K4 @ M2 )
=> ? [N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ( member_nat @ N3 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1264_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1265_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1266_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1267_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1268_infinite__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [R2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
& ! [N3: nat] : ( member_nat @ ( R2 @ N3 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_1269_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_1270_enumerate__Ex,axiom,
! [S: set_nat,S3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S3 @ S )
=> ? [N5: nat] :
( ( infini8530281810654367211te_nat @ S @ N5 )
= S3 ) ) ) ).
% enumerate_Ex
thf(fact_1271_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_1272_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1273_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
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y4: a] :
( ( if_a @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y4: a] :
( ( if_a @ $true @ X2 @ Y4 )
= X2 ) ).
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,
! [X2: nat,Y4: nat] :
( ( if_nat @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y4: nat] :
( ( if_nat @ $true @ X2 @ Y4 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_a @ ( elements_mat_a @ ( incide3343782147941204138of_b_a @ vs @ bs ) ) ).
%------------------------------------------------------------------------------