TPTP Problem File: SLH0092^1.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 : Finite_Fields/0005_Ring_Characteristic/prob_00205_006365__18159288_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1454 ( 391 unt; 182 typ; 0 def)
% Number of atoms : 4022 (1104 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11467 ( 227 ~; 24 |; 297 &;8913 @)
% ( 0 <=>;2006 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 1151 (1151 >; 0 *; 0 +; 0 <<)
% Number of symbols : 164 ( 161 usr; 22 con; 0-4 aty)
% Number of variables : 3672 ( 107 ^;3346 !; 219 ?;3672 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:21:12.597
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
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% Explicit typings (161)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Finite__Set_OFpow_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Int__Oint_J,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__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Int__Oint,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__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
the_in3678965191294008423nt_nat: set_set_int > ( set_int > nat ) > nat > set_int ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
the_in5441916425580749945et_int: set_set_int > ( set_int > set_int ) > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
uminus7346710233107665121et_int: set_set_int > set_set_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
semiring_1_Nats_int: set_int ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Nat__Onat,type,
semiring_1_Nats_nat: set_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_to2179722763343057421at_int: set_Pr7995236796853374141at_int ).
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__Set__Oset_It__Int__Oint_J_J,type,
top_top_set_set_int: set_set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mt__Int__Oint_J_J,type,
top_to6358659424274202653nt_int: set_Sum_sum_int_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J_J,type,
top_to8848742569205929409nt_nat: set_Sum_sum_int_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_to4171737849581180865at_int: set_Sum_sum_nat_int ).
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_Ring__Characteristic_Ofinite__field_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_f302724563095964181t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ofinite__field__axioms_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_f1119117527023254578t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OPow_001t__Int__Oint,type,
pow_int: set_int > set_set_int ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Int__Oint_J,type,
pow_set_int: set_set_int > set_set_set_int ).
thf(sy_c_Set_Obind_001t__Int__Oint_001t__Int__Oint,type,
bind_int_int: set_int > ( int > set_int ) > set_int ).
thf(sy_c_Set_Obind_001t__Int__Oint_001t__Nat__Onat,type,
bind_int_nat: set_int > ( int > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Int__Oint,type,
bind_nat_int: set_nat > ( nat > set_int ) > set_int ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: set_nat > ( nat > set_nat ) > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Int__Oint_M_Eo_J_001t__Set__Oset_It__Int__Oint_J,type,
image_int_o_set_int: ( ( int > $o ) > set_int ) > set_int_o > set_set_int ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
image_9165537771349138752et_int: ( ( set_int > $o ) > set_set_int ) > set_set_int_o > set_set_set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
image_int_set_int: ( int > set_int ) > set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
image_int_set_nat: ( int > set_nat ) > set_int > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
image_set_int_int: ( set_int > int ) > set_set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
image_set_int_nat: ( set_int > nat ) > set_set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_4702325430467532143et_nat: ( set_int > set_nat ) > set_set_int > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
image_1010086626112315521et_int: ( set_int > set_set_int ) > set_set_int > set_set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
image_4234937972324292645et_int: ( set_nat > set_set_int ) > set_set_nat > set_set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Int__Oint_J,type,
image_3513010637850279041et_int: ( set_set_int > set_int ) > set_set_set_int > set_set_int ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Int__Oint,type,
vimage_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Nat__Onat,type,
vimage_int_nat: ( int > nat ) > set_nat > set_int ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
vimage_int_set_int: ( int > set_int ) > set_set_int > set_int ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Int__Oint,type,
vimage_nat_int: ( nat > int ) > set_int > set_nat ).
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__Set__Oset_It__Int__Oint_J,type,
vimage_nat_set_int: ( nat > set_int ) > set_set_int > set_nat ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
vimage_set_int_int: ( set_int > int ) > set_int > set_set_int ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
vimage_set_int_nat: ( set_int > nat ) > set_nat > set_set_int ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
vimage6596094510776989313et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or5504389134266731388et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or458868116921152288et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or5935648273017318783et_int: set_int > set_set_int ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1268)
thf(fact_0__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_A_092_060Longrightarrow_062_Ax_A_092_060in_062_Azfact__iso_An_A_096_A_123_O_O_060n_125_092_060close_062,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ X @ ( image_nat_set_int @ ( ring_zfact_iso @ n ) @ ( set_ord_lessThan_nat @ n ) ) ) ) ).
% \<open>\<And>x. x \<in> carrier (ZFact (int n)) \<Longrightarrow> x \<in> zfact_iso n ` {..<n}\<close>
thf(fact_1__092_060open_062zfact__iso_An_A_096_A_123_O_O_060n_125_A_092_060subseteq_062_Acarrier_A_IZFact_A_Iint_An_J_J_092_060close_062,axiom,
ord_le4403425263959731960et_int @ ( image_nat_set_int @ ( ring_zfact_iso @ n ) @ ( set_ord_lessThan_nat @ n ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% \<open>zfact_iso n ` {..<n} \<subseteq> carrier (ZFact (int n))\<close>
thf(fact_2_zfact__iso__inj,axiom,
inj_on_nat_set_int @ ( ring_zfact_iso @ n ) @ ( set_ord_lessThan_nat @ n ) ).
% zfact_iso_inj
thf(fact_3_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_4_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_5_image__eqI,axiom,
! [B: set_int,F: set_int > set_int,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_6_image__eqI,axiom,
! [B: int,F: set_int > int,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_int @ B @ ( image_set_int_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_7_image__eqI,axiom,
! [B: nat,F: set_int > nat,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_nat @ B @ ( image_set_int_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_8_image__eqI,axiom,
! [B: set_int,F: int > set_int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_9_image__eqI,axiom,
! [B: int,F: int > int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B: nat,F: int > nat,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_nat @ B @ ( image_int_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: set_int,F: nat > set_int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_15_int__if,axiom,
! [P: $o,A3: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A3 @ B ) )
= ( semiri1314217659103216013at_int @ A3 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A3 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_16_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_17_Inf_OINF__cong,axiom,
! [A: set_set_int,B3: set_set_int,C: set_int > int,D: set_int > int,Inf: set_int > int] :
( ( A = B3 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_set_int_int @ C @ A ) )
= ( Inf @ ( image_set_int_int @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_18_Inf_OINF__cong,axiom,
! [A: set_int,B3: set_int,C: int > set_int,D: int > set_int,Inf: set_set_int > set_int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_set_int @ C @ A ) )
= ( Inf @ ( image_int_set_int @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_19_Inf_OINF__cong,axiom,
! [A: set_int,B3: set_int,C: int > nat,D: int > nat,Inf: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_nat @ C @ A ) )
= ( Inf @ ( image_int_nat @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_20_Inf_OINF__cong,axiom,
! [A: set_int,B3: set_int,C: int > int,D: int > int,Inf: set_int > int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_int @ C @ A ) )
= ( Inf @ ( image_int_int @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_21_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > int,D: nat > int,Inf: set_int > int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_int @ C @ A ) )
= ( Inf @ ( image_nat_int @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_22_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > set_int,D: nat > set_int,Inf: set_set_int > set_int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_set_int @ C @ A ) )
= ( Inf @ ( image_nat_set_int @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_23_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_24_Sup_OSUP__cong,axiom,
! [A: set_set_int,B3: set_set_int,C: set_int > int,D: set_int > int,Sup: set_int > int] :
( ( A = B3 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_set_int_int @ C @ A ) )
= ( Sup @ ( image_set_int_int @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_25_Sup_OSUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > set_int,D: int > set_int,Sup: set_set_int > set_int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_set_int @ C @ A ) )
= ( Sup @ ( image_int_set_int @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_26_Sup_OSUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > nat,D: int > nat,Sup: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_nat @ C @ A ) )
= ( Sup @ ( image_int_nat @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_27_Sup_OSUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > int,D: int > int,Sup: set_int > int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_int @ C @ A ) )
= ( Sup @ ( image_int_int @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_28_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > set_int,D: nat > set_int,Sup: set_set_int > set_int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_set_int @ C @ A ) )
= ( Sup @ ( image_nat_set_int @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_29_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_30_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > int,D: nat > int,Sup: set_int > int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_int @ C @ A ) )
= ( Sup @ ( image_nat_int @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_31_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_524474410958335435et_int @ F @ A ) ) ) ).
% imageI
thf(fact_32_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > int] :
( ( member_set_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_set_int_int @ F @ A ) ) ) ).
% imageI
thf(fact_33_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > nat] :
( ( member_set_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_set_int_nat @ F @ A ) ) ) ).
% imageI
thf(fact_34_imageI,axiom,
! [X: int,A: set_int,F: int > set_int] :
( ( member_int @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ A ) ) ) ).
% imageI
thf(fact_35_imageI,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_int_int @ F @ A ) ) ) ).
% imageI
thf(fact_36_imageI,axiom,
! [X: int,A: set_int,F: int > nat] :
( ( member_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ A ) ) ) ).
% imageI
thf(fact_37_imageI,axiom,
! [X: nat,A: set_nat,F: nat > set_int] :
( ( member_nat @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ A ) ) ) ).
% imageI
thf(fact_38_imageI,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A ) ) ) ).
% imageI
thf(fact_39_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_40_image__iff,axiom,
! [Z2: set_int,F: nat > set_int,A: set_nat] :
( ( member_set_int @ Z2 @ ( image_nat_set_int @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_41_image__iff,axiom,
! [Z2: set_int,F: int > set_int,A: set_int] :
( ( member_set_int @ Z2 @ ( image_int_set_int @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_42_image__iff,axiom,
! [Z2: int,F: set_int > int,A: set_set_int] :
( ( member_int @ Z2 @ ( image_set_int_int @ F @ A ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_43_image__iff,axiom,
! [Z2: int,F: nat > int,A: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_44_image__iff,axiom,
! [Z2: int,F: int > int,A: set_int] :
( ( member_int @ Z2 @ ( image_int_int @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_45_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_46_image__iff,axiom,
! [Z2: nat,F: int > nat,A: set_int] :
( ( member_nat @ Z2 @ ( image_int_nat @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_47_bex__imageD,axiom,
! [F: nat > set_int,A: set_nat,P: set_int > $o] :
( ? [X4: set_int] :
( ( member_set_int @ X4 @ ( image_nat_set_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_48_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_49_bex__imageD,axiom,
! [F: set_int > int,A: set_set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_set_int_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_50_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_51_bex__imageD,axiom,
! [F: int > set_int,A: set_int,P: set_int > $o] :
( ? [X4: set_int] :
( ( member_set_int @ X4 @ ( image_int_set_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_52_bex__imageD,axiom,
! [F: int > nat,A: set_int,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_int_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_53_bex__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_int_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_54_subsetI,axiom,
! [A: set_int,B3: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B3 ) )
=> ( ord_less_eq_set_int @ A @ B3 ) ) ).
% subsetI
thf(fact_55_subsetI,axiom,
! [A: set_set_int,B3: set_set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_set_int @ X2 @ B3 ) )
=> ( ord_le4403425263959731960et_int @ A @ B3 ) ) ).
% subsetI
thf(fact_56_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_57_subset__antisym,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_58_subset__antisym,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_59_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_60_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_61_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_62_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_63_verit__comp__simplify1_I2_J,axiom,
! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_64_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_65_verit__comp__simplify1_I2_J,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_66_verit__comp__simplify1_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_67_in__mono,axiom,
! [A: set_int,B3: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( member_int @ X @ A )
=> ( member_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_68_in__mono,axiom,
! [A: set_set_int,B3: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( member_set_int @ X @ A )
=> ( member_set_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_69_in__mono,axiom,
! [A: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_70_subsetD,axiom,
! [A: set_int,B3: set_int,C2: int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_71_subsetD,axiom,
! [A: set_set_int,B3: set_set_int,C2: set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( member_set_int @ C2 @ A )
=> ( member_set_int @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_72_subsetD,axiom,
! [A: set_nat,B3: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_73_equalityE,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( A = B3 )
=> ~ ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ~ ( ord_le4403425263959731960et_int @ B3 @ A ) ) ) ).
% equalityE
thf(fact_74_equalityE,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_75_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B4: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_int @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_76_subset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
! [X3: set_int] :
( ( member_set_int @ X3 @ A4 )
=> ( member_set_int @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_77_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_78_equalityD1,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( A = B3 )
=> ( ord_le4403425263959731960et_int @ A @ B3 ) ) ).
% equalityD1
thf(fact_79_equalityD1,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% equalityD1
thf(fact_80_equalityD2,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( A = B3 )
=> ( ord_le4403425263959731960et_int @ B3 @ A ) ) ).
% equalityD2
thf(fact_81_equalityD2,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% equalityD2
thf(fact_82_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B4: set_int] :
! [T: int] :
( ( member_int @ T @ A4 )
=> ( member_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_83_subset__iff,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
! [T: set_int] :
( ( member_set_int @ T @ A4 )
=> ( member_set_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_84_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_85_subset__refl,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ A @ A ) ).
% subset_refl
thf(fact_86_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_87_Collect__mono,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X2: set_int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_88_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_89_subset__trans,axiom,
! [A: set_set_int,B3: set_set_int,C: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C )
=> ( ord_le4403425263959731960et_int @ A @ C ) ) ) ).
% subset_trans
thf(fact_90_subset__trans,axiom,
! [A: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_91_set__eq__subset,axiom,
( ( ^ [Y2: set_set_int,Z: set_set_int] : ( Y2 = Z ) )
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_92_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_93_Collect__mono__iff,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) )
= ( ! [X3: set_int] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_94_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_95_verit__la__disequality,axiom,
! [A3: nat,B: nat] :
( ( A3 = B )
| ~ ( ord_less_eq_nat @ A3 @ B )
| ~ ( ord_less_eq_nat @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_96_verit__la__disequality,axiom,
! [A3: int,B: int] :
( ( A3 = B )
| ~ ( ord_less_eq_int @ A3 @ B )
| ~ ( ord_less_eq_int @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_97_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_98_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_99_inj__on__of__nat,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ semiri1316708129612266289at_nat @ N2 ) ).
% inj_on_of_nat
thf(fact_100_inj__on__of__nat,axiom,
! [N2: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N2 ) ).
% inj_on_of_nat
thf(fact_101_subset__image__iff,axiom,
! [B3: set_int,F: int > int,A: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B3
= ( image_int_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_102_subset__image__iff,axiom,
! [B3: set_int,F: set_int > int,A: set_set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_set_int_int @ F @ A ) )
= ( ? [AA: set_set_int] :
( ( ord_le4403425263959731960et_int @ AA @ A )
& ( B3
= ( image_set_int_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_103_subset__image__iff,axiom,
! [B3: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_104_subset__image__iff,axiom,
! [B3: set_set_int,F: int > set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_int_set_int @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B3
= ( image_int_set_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_105_subset__image__iff,axiom,
! [B3: set_set_int,F: set_int > set_int,A: set_set_int] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_524474410958335435et_int @ F @ A ) )
= ( ? [AA: set_set_int] :
( ( ord_le4403425263959731960et_int @ AA @ A )
& ( B3
= ( image_524474410958335435et_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_106_subset__image__iff,axiom,
! [B3: set_set_int,F: nat > set_int,A: set_nat] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_nat_set_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_set_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_107_subset__image__iff,axiom,
! [B3: set_nat,F: int > nat,A: set_int] :
( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A )
& ( B3
= ( image_int_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_108_subset__image__iff,axiom,
! [B3: set_nat,F: set_int > nat,A: set_set_int] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_int_nat @ F @ A ) )
= ( ? [AA: set_set_int] :
( ( ord_le4403425263959731960et_int @ AA @ A )
& ( B3
= ( image_set_int_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_109_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_110_image__subset__iff,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_111_image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_112_image__subset__iff,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_113_image__subset__iff,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_int @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_114_image__subset__iff,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_set_int @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_115_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_116_image__subset__iff,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_117_subset__imageE,axiom,
! [B3: set_int,F: int > int,A: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B3
!= ( image_int_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_118_subset__imageE,axiom,
! [B3: set_int,F: set_int > int,A: set_set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_set_int_int @ F @ A ) )
=> ~ ! [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
=> ( B3
!= ( image_set_int_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_119_subset__imageE,axiom,
! [B3: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B3
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_120_subset__imageE,axiom,
! [B3: set_set_int,F: int > set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_int_set_int @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B3
!= ( image_int_set_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_121_subset__imageE,axiom,
! [B3: set_set_int,F: set_int > set_int,A: set_set_int] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_524474410958335435et_int @ F @ A ) )
=> ~ ! [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
=> ( B3
!= ( image_524474410958335435et_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_122_subset__imageE,axiom,
! [B3: set_set_int,F: nat > set_int,A: set_nat] :
( ( ord_le4403425263959731960et_int @ B3 @ ( image_nat_set_int @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B3
!= ( image_nat_set_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_123_subset__imageE,axiom,
! [B3: set_nat,F: int > nat,A: set_int] :
( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A ) )
=> ~ ! [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
=> ( B3
!= ( image_int_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_124_subset__imageE,axiom,
! [B3: set_nat,F: set_int > nat,A: set_set_int] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_int_nat @ F @ A ) )
=> ~ ! [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
=> ( B3
!= ( image_set_int_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_125_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B3
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_126_image__subsetI,axiom,
! [A: set_set_int,F: set_int > int,B3: set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_127_image__subsetI,axiom,
! [A: set_int,F: int > int,B3: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_128_image__subsetI,axiom,
! [A: set_nat,F: nat > int,B3: set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_129_image__subsetI,axiom,
! [A: set_set_int,F: set_int > set_int,B3: set_set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_130_image__subsetI,axiom,
! [A: set_int,F: int > set_int,B3: set_set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_131_image__subsetI,axiom,
! [A: set_nat,F: nat > set_int,B3: set_set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B3 ) )
=> ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_132_image__subsetI,axiom,
! [A: set_set_int,F: set_int > nat,B3: set_nat] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_set_int_nat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_133_image__subsetI,axiom,
! [A: set_int,F: int > nat,B3: set_nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_134_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_135_image__mono,axiom,
! [A: set_int,B3: set_int,F: int > int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ ( image_int_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_136_image__mono,axiom,
! [A: set_int,B3: set_int,F: int > set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ ( image_int_set_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_137_image__mono,axiom,
! [A: set_int,B3: set_int,F: int > nat] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ ( image_int_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_138_image__mono,axiom,
! [A: set_set_int,B3: set_set_int,F: set_int > int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ ( image_set_int_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_139_image__mono,axiom,
! [A: set_set_int,B3: set_set_int,F: set_int > set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A ) @ ( image_524474410958335435et_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_140_image__mono,axiom,
! [A: set_set_int,B3: set_set_int,F: set_int > nat] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( image_set_int_nat @ F @ A ) @ ( image_set_int_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_141_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_142_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > set_int] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ ( image_nat_set_int @ F @ B3 ) ) ) ).
% image_mono
thf(fact_143_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_144_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_145_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: int,F: set_int > int] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_set_int_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_146_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: nat,F: set_int > nat] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_set_int_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_147_rev__image__eqI,axiom,
! [X: int,A: set_int,B: set_int,F: int > set_int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_148_rev__image__eqI,axiom,
! [X: int,A: set_int,B: int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_149_rev__image__eqI,axiom,
! [X: int,A: set_int,B: nat,F: int > nat] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_150_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: set_int,F: nat > set_int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_151_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_152_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_153_ball__imageD,axiom,
! [F: nat > set_int,A: set_nat,P: set_int > $o] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( image_nat_set_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_154_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_155_ball__imageD,axiom,
! [F: set_int > int,A: set_set_int,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_set_int_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: set_int] :
( ( member_set_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_156_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_157_ball__imageD,axiom,
! [F: int > set_int,A: set_int,P: set_int > $o] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( image_int_set_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_158_ball__imageD,axiom,
! [F: int > nat,A: set_int,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_159_ball__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_160_image__cong,axiom,
! [M2: set_set_int,N2: set_set_int,F: set_int > int,G: set_int > int] :
( ( M2 = N2 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_set_int_int @ F @ M2 )
= ( image_set_int_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_161_image__cong,axiom,
! [M2: set_int,N2: set_int,F: int > set_int,G: int > set_int] :
( ( M2 = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_set_int @ F @ M2 )
= ( image_int_set_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_162_image__cong,axiom,
! [M2: set_int,N2: set_int,F: int > nat,G: int > nat] :
( ( M2 = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_nat @ F @ M2 )
= ( image_int_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_163_image__cong,axiom,
! [M2: set_int,N2: set_int,F: int > int,G: int > int] :
( ( M2 = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_int @ F @ M2 )
= ( image_int_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_164_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > set_int,G: nat > set_int] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_set_int @ F @ M2 )
= ( image_nat_set_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_165_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_166_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > int,G: nat > int] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_int @ F @ M2 )
= ( image_nat_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_167_subset__image__inj,axiom,
! [S: set_int,F: int > int,T2: set_int] :
( ( ord_less_eq_set_int @ S @ ( image_int_int @ F @ T2 ) )
= ( ? [U: set_int] :
( ( ord_less_eq_set_int @ U @ T2 )
& ( inj_on_int_int @ F @ U )
& ( S
= ( image_int_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_168_subset__image__inj,axiom,
! [S: set_int,F: set_int > int,T2: set_set_int] :
( ( ord_less_eq_set_int @ S @ ( image_set_int_int @ F @ T2 ) )
= ( ? [U: set_set_int] :
( ( ord_le4403425263959731960et_int @ U @ T2 )
& ( inj_on_set_int_int @ F @ U )
& ( S
= ( image_set_int_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_169_subset__image__inj,axiom,
! [S: set_int,F: nat > int,T2: set_nat] :
( ( ord_less_eq_set_int @ S @ ( image_nat_int @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_int @ F @ U )
& ( S
= ( image_nat_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_170_subset__image__inj,axiom,
! [S: set_set_int,F: int > set_int,T2: set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( image_int_set_int @ F @ T2 ) )
= ( ? [U: set_int] :
( ( ord_less_eq_set_int @ U @ T2 )
& ( inj_on_int_set_int @ F @ U )
& ( S
= ( image_int_set_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_171_subset__image__inj,axiom,
! [S: set_set_int,F: set_int > set_int,T2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( image_524474410958335435et_int @ F @ T2 ) )
= ( ? [U: set_set_int] :
( ( ord_le4403425263959731960et_int @ U @ T2 )
& ( inj_on6435365835345961783et_int @ F @ U )
& ( S
= ( image_524474410958335435et_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_172_subset__image__inj,axiom,
! [S: set_set_int,F: nat > set_int,T2: set_nat] :
( ( ord_le4403425263959731960et_int @ S @ ( image_nat_set_int @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_set_int @ F @ U )
& ( S
= ( image_nat_set_int @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_173_subset__image__inj,axiom,
! [S: set_nat,F: int > nat,T2: set_int] :
( ( ord_less_eq_set_nat @ S @ ( image_int_nat @ F @ T2 ) )
= ( ? [U: set_int] :
( ( ord_less_eq_set_int @ U @ T2 )
& ( inj_on_int_nat @ F @ U )
& ( S
= ( image_int_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_174_subset__image__inj,axiom,
! [S: set_nat,F: set_int > nat,T2: set_set_int] :
( ( ord_less_eq_set_nat @ S @ ( image_set_int_nat @ F @ T2 ) )
= ( ? [U: set_set_int] :
( ( ord_le4403425263959731960et_int @ U @ T2 )
& ( inj_on_set_int_nat @ F @ U )
& ( S
= ( image_set_int_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_175_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_nat @ F @ U )
& ( S
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_176_inj__on__image__mem__iff,axiom,
! [F: int > set_int,B3: set_int,A3: int,A: set_int] :
( ( inj_on_int_set_int @ F @ B3 )
=> ( ( member_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_int_set_int @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_177_inj__on__image__mem__iff,axiom,
! [F: int > int,B3: set_int,A3: int,A: set_int] :
( ( inj_on_int_int @ F @ B3 )
=> ( ( member_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( member_int @ ( F @ A3 ) @ ( image_int_int @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_178_inj__on__image__mem__iff,axiom,
! [F: int > nat,B3: set_int,A3: int,A: set_int] :
( ( inj_on_int_nat @ F @ B3 )
=> ( ( member_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_int_nat @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_179_inj__on__image__mem__iff,axiom,
! [F: set_int > set_int,B3: set_set_int,A3: set_int,A: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ B3 )
=> ( ( member_set_int @ A3 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_524474410958335435et_int @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_180_inj__on__image__mem__iff,axiom,
! [F: set_int > int,B3: set_set_int,A3: set_int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ B3 )
=> ( ( member_set_int @ A3 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( member_int @ ( F @ A3 ) @ ( image_set_int_int @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_181_inj__on__image__mem__iff,axiom,
! [F: set_int > nat,B3: set_set_int,A3: set_int,A: set_set_int] :
( ( inj_on_set_int_nat @ F @ B3 )
=> ( ( member_set_int @ A3 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_set_int_nat @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_182_inj__on__image__mem__iff,axiom,
! [F: nat > set_int,B3: set_nat,A3: nat,A: set_nat] :
( ( inj_on_nat_set_int @ F @ B3 )
=> ( ( member_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_nat_set_int @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_183_inj__on__image__mem__iff,axiom,
! [F: nat > int,B3: set_nat,A3: nat,A: set_nat] :
( ( inj_on_nat_int @ F @ B3 )
=> ( ( member_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_int @ ( F @ A3 ) @ ( image_nat_int @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_184_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B3: set_nat,A3: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B3 )
=> ( ( member_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_185_inj__on__image__eq__iff,axiom,
! [F: int > set_int,C: set_int,A: set_int,B3: set_int] :
( ( inj_on_int_set_int @ F @ C )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ( ( image_int_set_int @ F @ A )
= ( image_int_set_int @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_186_inj__on__image__eq__iff,axiom,
! [F: int > nat,C: set_int,A: set_int,B3: set_int] :
( ( inj_on_int_nat @ F @ C )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ( ( image_int_nat @ F @ A )
= ( image_int_nat @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_187_inj__on__image__eq__iff,axiom,
! [F: int > int,C: set_int,A: set_int,B3: set_int] :
( ( inj_on_int_int @ F @ C )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( ord_less_eq_set_int @ B3 @ C )
=> ( ( ( image_int_int @ F @ A )
= ( image_int_int @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_188_inj__on__image__eq__iff,axiom,
! [F: set_int > int,C: set_set_int,A: set_set_int,B3: set_set_int] :
( ( inj_on_set_int_int @ F @ C )
=> ( ( ord_le4403425263959731960et_int @ A @ C )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C )
=> ( ( ( image_set_int_int @ F @ A )
= ( image_set_int_int @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_189_inj__on__image__eq__iff,axiom,
! [F: nat > int,C: set_nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_int @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ( ( image_nat_int @ F @ A )
= ( image_nat_int @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_190_inj__on__image__eq__iff,axiom,
! [F: nat > set_int,C: set_nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_set_int @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ( ( image_nat_set_int @ F @ A )
= ( image_nat_set_int @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_191_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B3 ) )
= ( A = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_192_mem__Collect__eq,axiom,
! [A3: set_int,P: set_int > $o] :
( ( member_set_int @ A3 @ ( collect_set_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_195_Collect__mem__eq,axiom,
! [A: set_set_int] :
( ( collect_set_int
@ ^ [X3: set_int] : ( member_set_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_198_order__refl,axiom,
! [X: set_set_int] : ( ord_le4403425263959731960et_int @ X @ X ) ).
% order_refl
thf(fact_199_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_200_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_201_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_202_dual__order_Orefl,axiom,
! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_203_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_204_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_205_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_206_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > set_int,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_set_int @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_set_int @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_207_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_208_inj__on__subset,axiom,
! [F: nat > set_int,A: set_nat,B3: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( inj_on_nat_set_int @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_209_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( inj_on_nat_nat @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_210_subset__inj__on,axiom,
! [F: nat > set_int,B3: set_nat,A: set_nat] :
( ( inj_on_nat_set_int @ F @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( inj_on_nat_set_int @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_211_subset__inj__on,axiom,
! [F: nat > nat,B3: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_212_all__subset__image,axiom,
! [F: int > int,A: set_int,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A )
=> ( P @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_213_all__subset__image,axiom,
! [F: set_int > int,A: set_set_int,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ ( image_set_int_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ A )
=> ( P @ ( image_set_int_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_214_all__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_215_all__subset__image,axiom,
! [F: int > set_int,A: set_int,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ ( image_int_set_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A )
=> ( P @ ( image_int_set_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_216_all__subset__image,axiom,
! [F: set_int > set_int,A: set_set_int,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ ( image_524474410958335435et_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ A )
=> ( P @ ( image_524474410958335435et_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_217_all__subset__image,axiom,
! [F: nat > set_int,A: set_nat,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ ( image_nat_set_int @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_set_int @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_218_all__subset__image,axiom,
! [F: int > nat,A: set_int,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A )
=> ( P @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_219_all__subset__image,axiom,
! [F: set_int > nat,A: set_set_int,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_set_int_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ A )
=> ( P @ ( image_set_int_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_220_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_221_inj__onD,axiom,
! [F: nat > set_int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_222_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_223_inj__onI,axiom,
! [A: set_nat,F: nat > set_int] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on_nat_set_int @ F @ A ) ) ).
% inj_onI
thf(fact_224_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_225_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_226_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_227_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_228_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_229_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_230_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_231_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_232_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_233_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_234_order__antisym__conv,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ Y @ X )
=> ( ( ord_le4403425263959731960et_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_235_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_236_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_237_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_238_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_239_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_240_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_241_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_242_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_243_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__le__eq__subst,axiom,
! [A3: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_247_ord__le__eq__subst,axiom,
! [A3: set_nat,B: set_nat,F: set_nat > int,C2: int] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A3: set_set_int,B: set_set_int,F: set_set_int > nat,C2: nat] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A3: set_set_int,B: set_set_int,F: set_set_int > int,C2: int] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_251_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A3: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A3: set_nat,F: int > set_nat,B: int,C2: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__eq__le__subst,axiom,
! [A3: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
! [A3: int,F: set_nat > int,B: set_nat,C2: set_nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_258_ord__eq__le__subst,axiom,
! [A3: nat,F: set_set_int > nat,B: set_set_int,C2: set_set_int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_le4403425263959731960et_int @ B @ C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_259_ord__eq__le__subst,axiom,
! [A3: int,F: set_set_int > int,B: set_set_int,C2: set_set_int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_le4403425263959731960et_int @ B @ C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_260_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_261_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_262_order__eq__refl,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( X = Y )
=> ( ord_le4403425263959731960et_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_263_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_264_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_265_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_266_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_267_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_268_order__subst2,axiom,
! [A3: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_269_order__subst2,axiom,
! [A3: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_270_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_271_order__subst2,axiom,
! [A3: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_272_order__subst2,axiom,
! [A3: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_273_order__subst2,axiom,
! [A3: set_nat,B: set_nat,F: set_nat > int,C2: int] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_274_order__subst2,axiom,
! [A3: set_set_int,B: set_set_int,F: set_set_int > nat,C2: nat] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_275_order__subst2,axiom,
! [A3: set_set_int,B: set_set_int,F: set_set_int > int,C2: int] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_276_order__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_277_order__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_278_order__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_279_order__subst1,axiom,
! [A3: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_280_order__subst1,axiom,
! [A3: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_281_order__subst1,axiom,
! [A3: int,F: set_nat > int,B: set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_282_order__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_283_order__subst1,axiom,
! [A3: set_nat,F: int > set_nat,B: int,C2: int] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_284_order__subst1,axiom,
! [A3: set_set_int,F: nat > set_set_int,B: nat,C2: nat] :
( ( ord_le4403425263959731960et_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4403425263959731960et_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4403425263959731960et_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_285_order__subst1,axiom,
! [A3: set_set_int,F: int > set_set_int,B: int,C2: int] :
( ( ord_le4403425263959731960et_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_le4403425263959731960et_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4403425263959731960et_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_286_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_int,Z: set_set_int] : ( Y2 = Z ) )
= ( ^ [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
& ( ord_le4403425263959731960et_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_287_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_288_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_289_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_290_antisym,axiom,
! [A3: set_set_int,B: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ord_le4403425263959731960et_int @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_291_antisym,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_292_antisym,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_293_antisym,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_294_dual__order_Otrans,axiom,
! [B: set_set_int,A3: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ B @ A3 )
=> ( ( ord_le4403425263959731960et_int @ C2 @ B )
=> ( ord_le4403425263959731960et_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_295_dual__order_Otrans,axiom,
! [B: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_296_dual__order_Otrans,axiom,
! [B: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_297_dual__order_Otrans,axiom,
! [B: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_298_dual__order_Oantisym,axiom,
! [B: set_set_int,A3: set_set_int] :
( ( ord_le4403425263959731960et_int @ B @ A3 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_299_dual__order_Oantisym,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_300_dual__order_Oantisym,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_301_dual__order_Oantisym,axiom,
! [B: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_302_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_int,Z: set_set_int] : ( Y2 = Z ) )
= ( ^ [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ B2 @ A2 )
& ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_303_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_304_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_305_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_306_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_307_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_308_order__trans,axiom,
! [X: set_set_int,Y: set_set_int,Z2: set_set_int] :
( ( ord_le4403425263959731960et_int @ X @ Y )
=> ( ( ord_le4403425263959731960et_int @ Y @ Z2 )
=> ( ord_le4403425263959731960et_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_309_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_310_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_311_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_312_order_Otrans,axiom,
! [A3: set_set_int,B: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( ord_le4403425263959731960et_int @ B @ C2 )
=> ( ord_le4403425263959731960et_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_313_order_Otrans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_314_order_Otrans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_315_order_Otrans,axiom,
! [A3: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_316_order__antisym,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ X @ Y )
=> ( ( ord_le4403425263959731960et_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_317_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_318_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_319_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_320_ord__le__eq__trans,axiom,
! [A3: set_set_int,B: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_le4403425263959731960et_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_321_ord__le__eq__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_322_ord__le__eq__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_323_ord__le__eq__trans,axiom,
! [A3: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_324_ord__eq__le__trans,axiom,
! [A3: set_set_int,B: set_set_int,C2: set_set_int] :
( ( A3 = B )
=> ( ( ord_le4403425263959731960et_int @ B @ C2 )
=> ( ord_le4403425263959731960et_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_325_ord__eq__le__trans,axiom,
! [A3: nat,B: nat,C2: nat] :
( ( A3 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_326_ord__eq__le__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( A3 = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_327_ord__eq__le__trans,axiom,
! [A3: set_nat,B: set_nat,C2: set_nat] :
( ( A3 = B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_328_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_int,Z: set_set_int] : ( Y2 = Z ) )
= ( ^ [X3: set_set_int,Y5: set_set_int] :
( ( ord_le4403425263959731960et_int @ X3 @ Y5 )
& ( ord_le4403425263959731960et_int @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_329_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_330_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_331_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_332_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_333_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_334_nle__le,axiom,
! [A3: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B ) )
= ( ( ord_less_eq_nat @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_335_nle__le,axiom,
! [A3: int,B: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B ) )
= ( ( ord_less_eq_int @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_336_inj__on__inverseI,axiom,
! [A: set_nat,G: set_int > nat,F: nat > set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_nat_set_int @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_337_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_338_inj__on__contraD,axiom,
! [F: nat > set_int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_339_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_340_inj__on__eq__iff,axiom,
! [F: nat > set_int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_341_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_342_inj__on__cong,axiom,
! [A: set_nat,F: nat > set_int,G: nat > set_int] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_set_int @ F @ A )
= ( inj_on_nat_set_int @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_343_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_344_inj__on__def,axiom,
( inj_on_nat_set_int
= ( ^ [F2: nat > set_int,A4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_345_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_346_image__Fpow__mono,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_3513010637850279041et_int @ ( image_set_int_int @ F ) @ ( finite_Fpow_set_int @ A ) ) @ ( finite_Fpow_int @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_347_image__Fpow__mono,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_int @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_348_image__Fpow__mono,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A ) ) @ ( finite_Fpow_int @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_349_image__Fpow__mono,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
=> ( ord_le4317611570275147438et_int @ ( image_4234937972324292645et_int @ ( image_nat_set_int @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_set_int @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_350_image__Fpow__mono,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
=> ( ord_le4317611570275147438et_int @ ( image_1010086626112315521et_int @ ( image_int_set_int @ F ) @ ( finite_Fpow_int @ A ) ) @ ( finite_Fpow_set_int @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_351_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_352_image__Fpow__mono,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( finite_Fpow_int @ A ) ) @ ( finite_Fpow_nat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_353_the__inv__into__into,axiom,
! [F: int > set_int,A: set_int,X: set_int,B3: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( member_set_int @ X @ ( image_int_set_int @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( member_int @ ( the_in1251481061984003779et_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_354_the__inv__into__into,axiom,
! [F: int > int,A: set_int,X: int,B3: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( member_int @ X @ ( image_int_int @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( member_int @ ( the_inv_into_int_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_355_the__inv__into__into,axiom,
! [F: int > nat,A: set_int,X: nat,B3: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_int_nat @ F @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( member_int @ ( the_inv_into_int_nat @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_356_the__inv__into__into,axiom,
! [F: set_int > set_int,A: set_set_int,X: set_int,B3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ A )
=> ( ( member_set_int @ X @ ( image_524474410958335435et_int @ F @ A ) )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( member_set_int @ ( the_in5441916425580749945et_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_357_the__inv__into__into,axiom,
! [F: set_int > int,A: set_set_int,X: int,B3: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( member_int @ X @ ( image_set_int_int @ F @ A ) )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( member_set_int @ ( the_in3676474720784958147nt_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_358_the__inv__into__into,axiom,
! [F: set_int > nat,A: set_set_int,X: nat,B3: set_set_int] :
( ( inj_on_set_int_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_set_int_nat @ F @ A ) )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( member_set_int @ ( the_in3678965191294008423nt_nat @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_359_the__inv__into__into,axiom,
! [F: nat > set_int,A: set_nat,X: set_int,B3: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( member_set_int @ X @ ( image_nat_set_int @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( member_nat @ ( the_in879827501747159143et_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_360_the__inv__into__into,axiom,
! [F: nat > int,A: set_nat,X: int,B3: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( member_int @ X @ ( image_nat_int @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( member_nat @ ( the_inv_into_nat_int @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_361_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X: nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_362_Fpow__mono,axiom,
! [A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( finite_Fpow_int @ A ) @ ( finite_Fpow_int @ B3 ) ) ) ).
% Fpow_mono
thf(fact_363_Fpow__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_le4317611570275147438et_int @ ( finite_Fpow_set_int @ A ) @ ( finite_Fpow_set_int @ B3 ) ) ) ).
% Fpow_mono
thf(fact_364_Fpow__mono,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B3 ) ) ) ).
% Fpow_mono
thf(fact_365_the__inv__into__onto,axiom,
! [F: set_int > nat,A: set_set_int] :
( ( inj_on_set_int_nat @ F @ A )
=> ( ( image_nat_set_int @ ( the_in3678965191294008423nt_nat @ A @ F ) @ ( image_set_int_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_366_the__inv__into__onto,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( image_set_int_int @ ( the_in1251481061984003779et_int @ A @ F ) @ ( image_int_set_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_367_the__inv__into__onto,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( image_nat_int @ ( the_inv_into_int_nat @ A @ F ) @ ( image_int_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_368_the__inv__into__onto,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( image_int_set_int @ ( the_in3676474720784958147nt_int @ A @ F ) @ ( image_set_int_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_369_the__inv__into__onto,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( image_int_nat @ ( the_inv_into_nat_int @ A @ F ) @ ( image_nat_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_370_the__inv__into__onto,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( image_int_int @ ( the_inv_into_int_int @ A @ F ) @ ( image_int_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_371_the__inv__into__onto,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( image_set_int_nat @ ( the_in879827501747159143et_int @ A @ F ) @ ( image_nat_set_int @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_372_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_373_inj__on__image__Fpow,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( inj_on8788328216580801005et_int @ ( image_set_int_int @ F ) @ ( finite_Fpow_set_int @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_374_inj__on__image__Fpow,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_375_inj__on__image__Fpow,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( inj_on6285404204842837485et_int @ ( image_int_set_int @ F ) @ ( finite_Fpow_int @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_376_inj__on__image__Fpow,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ ( finite_Fpow_int @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_377_inj__on__image__Fpow,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_378_inj__on__image__Fpow,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( inj_on286883514200038801et_int @ ( image_nat_set_int @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_379_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_380_Greatest__equality,axiom,
! [P: set_set_int > $o,X: set_set_int] :
( ( P @ X )
=> ( ! [Y3: set_set_int] :
( ( P @ Y3 )
=> ( ord_le4403425263959731960et_int @ Y3 @ X ) )
=> ( ( order_8012710290990177407et_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_381_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_382_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_383_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_384_GreatestI2__order,axiom,
! [P: set_set_int > $o,X: set_set_int,Q: set_set_int > $o] :
( ( P @ X )
=> ( ! [Y3: set_set_int] :
( ( P @ Y3 )
=> ( ord_le4403425263959731960et_int @ Y3 @ X ) )
=> ( ! [X2: set_set_int] :
( ( P @ X2 )
=> ( ! [Y4: set_set_int] :
( ( P @ Y4 )
=> ( ord_le4403425263959731960et_int @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_8012710290990177407et_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_385_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( ! [Y4: int] :
( ( P @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_386_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_387_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_388_image__Pow__mono,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_3513010637850279041et_int @ ( image_set_int_int @ F ) @ ( pow_set_int @ A ) ) @ ( pow_int @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_389_image__Pow__mono,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A ) ) @ ( pow_int @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_390_image__Pow__mono,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( pow_int @ A ) ) @ ( pow_int @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_391_image__Pow__mono,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
=> ( ord_le4317611570275147438et_int @ ( image_4234937972324292645et_int @ ( image_nat_set_int @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_int @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_392_image__Pow__mono,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
=> ( ord_le4317611570275147438et_int @ ( image_1010086626112315521et_int @ ( image_int_set_int @ F ) @ ( pow_int @ A ) ) @ ( pow_set_int @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_393_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_394_image__Pow__mono,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A ) ) @ ( pow_nat @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_395_inj__image__subset__iff,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ ( image_int_int @ F @ B3 ) )
= ( ord_less_eq_set_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_396_inj__image__subset__iff,axiom,
! [F: set_int > int,A: set_set_int,B3: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ ( image_set_int_int @ F @ B3 ) )
= ( ord_le4403425263959731960et_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_397_inj__image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B3: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B3 ) )
= ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_398_inj__image__subset__iff,axiom,
! [F: int > set_int,A: set_int,B3: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ ( image_int_set_int @ F @ B3 ) )
= ( ord_less_eq_set_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_399_inj__image__subset__iff,axiom,
! [F: set_int > set_int,A: set_set_int,B3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A ) @ ( image_524474410958335435et_int @ F @ B3 ) )
= ( ord_le4403425263959731960et_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_400_inj__image__subset__iff,axiom,
! [F: nat > set_int,A: set_nat,B3: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ ( image_nat_set_int @ F @ B3 ) )
= ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_401_inj__image__subset__iff,axiom,
! [F: int > nat,A: set_int,B3: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ ( image_int_nat @ F @ B3 ) )
= ( ord_less_eq_set_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_402_inj__image__subset__iff,axiom,
! [F: set_int > nat,A: set_set_int,B3: set_set_int] :
( ( inj_on_set_int_nat @ F @ top_top_set_set_int )
=> ( ( ord_less_eq_set_nat @ ( image_set_int_nat @ F @ A ) @ ( image_set_int_nat @ F @ B3 ) )
= ( ord_le4403425263959731960et_int @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_403_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B3 ) )
= ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_404_UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% UNIV_I
thf(fact_405_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_406_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_407_Pow__UNIV,axiom,
( ( pow_int @ top_top_set_int )
= top_top_set_set_int ) ).
% Pow_UNIV
thf(fact_408_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_409_PowI,axiom,
! [A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( member_set_int @ A @ ( pow_int @ B3 ) ) ) ).
% PowI
thf(fact_410_PowI,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( member_set_set_int @ A @ ( pow_set_int @ B3 ) ) ) ).
% PowI
thf(fact_411_PowI,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( member_set_nat @ A @ ( pow_nat @ B3 ) ) ) ).
% PowI
thf(fact_412_Pow__iff,axiom,
! [A: set_int,B3: set_int] :
( ( member_set_int @ A @ ( pow_int @ B3 ) )
= ( ord_less_eq_set_int @ A @ B3 ) ) ).
% Pow_iff
thf(fact_413_Pow__iff,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( member_set_set_int @ A @ ( pow_set_int @ B3 ) )
= ( ord_le4403425263959731960et_int @ A @ B3 ) ) ).
% Pow_iff
thf(fact_414_Pow__iff,axiom,
! [A: set_nat,B3: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B3 ) )
= ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% Pow_iff
thf(fact_415_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq_int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq_int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_416_UNIV__witness,axiom,
? [X2: set_int] : ( member_set_int @ X2 @ top_top_set_set_int ) ).
% UNIV_witness
thf(fact_417_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_418_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_419_UNIV__eq__I,axiom,
! [A: set_set_int] :
( ! [X2: set_int] : ( member_set_int @ X2 @ A )
=> ( top_top_set_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_420_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_421_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_422_Pow__top,axiom,
! [A: set_int] : ( member_set_int @ A @ ( pow_int @ A ) ) ).
% Pow_top
thf(fact_423_the__inv__f__f,axiom,
! [F: nat > set_int,X: nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( the_in879827501747159143et_int @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_424_the__inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_425_Fpow__subset__Pow,axiom,
! [A: set_int] : ( ord_le4403425263959731960et_int @ ( finite_Fpow_int @ A ) @ ( pow_int @ A ) ) ).
% Fpow_subset_Pow
thf(fact_426_top__greatest,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ top_top_set_int ) ).
% top_greatest
thf(fact_427_top__greatest,axiom,
! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ A3 @ top_top_set_set_int ) ).
% top_greatest
thf(fact_428_top__greatest,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% top_greatest
thf(fact_429_top_Oextremum__unique,axiom,
! [A3: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A3 )
= ( A3 = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_430_top_Oextremum__unique,axiom,
! [A3: set_set_int] :
( ( ord_le4403425263959731960et_int @ top_top_set_set_int @ A3 )
= ( A3 = top_top_set_set_int ) ) ).
% top.extremum_unique
thf(fact_431_top_Oextremum__unique,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A3 )
= ( A3 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_432_top_Oextremum__uniqueI,axiom,
! [A3: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A3 )
=> ( A3 = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_433_top_Oextremum__uniqueI,axiom,
! [A3: set_set_int] :
( ( ord_le4403425263959731960et_int @ top_top_set_set_int @ A3 )
=> ( A3 = top_top_set_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_434_top_Oextremum__uniqueI,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A3 )
=> ( A3 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_435_PowD,axiom,
! [A: set_int,B3: set_int] :
( ( member_set_int @ A @ ( pow_int @ B3 ) )
=> ( ord_less_eq_set_int @ A @ B3 ) ) ).
% PowD
thf(fact_436_PowD,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( member_set_set_int @ A @ ( pow_set_int @ B3 ) )
=> ( ord_le4403425263959731960et_int @ A @ B3 ) ) ).
% PowD
thf(fact_437_PowD,axiom,
! [A: set_nat,B3: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% PowD
thf(fact_438_image__Pow__surj,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( ( image_nat_set_int @ F @ A )
= B3 )
=> ( ( image_4234937972324292645et_int @ ( image_nat_set_int @ F ) @ ( pow_nat @ A ) )
= ( pow_set_int @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_439_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B3 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_440_image__Pow__surj,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( ( image_set_int_int @ F @ A )
= B3 )
=> ( ( image_3513010637850279041et_int @ ( image_set_int_int @ F ) @ ( pow_set_int @ A ) )
= ( pow_int @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_441_image__Pow__surj,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( ( image_nat_int @ F @ A )
= B3 )
=> ( ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A ) )
= ( pow_int @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_442_image__Pow__surj,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( ( image_int_set_int @ F @ A )
= B3 )
=> ( ( image_1010086626112315521et_int @ ( image_int_set_int @ F ) @ ( pow_int @ A ) )
= ( pow_set_int @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_443_image__Pow__surj,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( ( image_int_nat @ F @ A )
= B3 )
=> ( ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A ) )
= ( pow_nat @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_444_image__Pow__surj,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( ( image_int_int @ F @ A )
= B3 )
=> ( ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( pow_int @ A ) )
= ( pow_int @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_445_subset__UNIV,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% subset_UNIV
thf(fact_446_subset__UNIV,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ A @ top_top_set_set_int ) ).
% subset_UNIV
thf(fact_447_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_448_range__eqI,axiom,
! [B: int,F: set_int > int,X: set_int] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_set_int_int @ F @ top_top_set_set_int ) ) ) ).
% range_eqI
thf(fact_449_range__eqI,axiom,
! [B: set_int,F: int > set_int,X: int] :
( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_450_range__eqI,axiom,
! [B: int,F: int > int,X: int] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_451_range__eqI,axiom,
! [B: nat,F: int > nat,X: int] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_452_range__eqI,axiom,
! [B: set_int,F: nat > set_int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_453_range__eqI,axiom,
! [B: int,F: nat > int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_454_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_455_rangeI,axiom,
! [F: set_int > int,X: set_int] : ( member_int @ ( F @ X ) @ ( image_set_int_int @ F @ top_top_set_set_int ) ) ).
% rangeI
thf(fact_456_rangeI,axiom,
! [F: int > set_int,X: int] : ( member_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_457_rangeI,axiom,
! [F: int > int,X: int] : ( member_int @ ( F @ X ) @ ( image_int_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_458_rangeI,axiom,
! [F: int > nat,X: int] : ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_459_rangeI,axiom,
! [F: nat > set_int,X: nat] : ( member_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_460_rangeI,axiom,
! [F: nat > int,X: nat] : ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_461_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_462_surj__def,axiom,
! [F: set_int > int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: set_int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_463_surj__def,axiom,
! [F: int > set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
= ( ! [Y5: set_int] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_464_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_465_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_466_surj__def,axiom,
! [F: nat > set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
= ( ! [Y5: set_int] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_467_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_468_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_469_surjI,axiom,
! [G: set_int > int,F: int > set_int] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_set_int_int @ G @ top_top_set_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_470_surjI,axiom,
! [G: int > set_int,F: set_int > int] :
( ! [X2: set_int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_set_int @ G @ top_top_set_int )
= top_top_set_set_int ) ) ).
% surjI
thf(fact_471_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_472_surjI,axiom,
! [G: int > nat,F: nat > int] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_nat @ G @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_473_surjI,axiom,
! [G: nat > set_int,F: set_int > nat] :
( ! [X2: set_int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_set_int @ G @ top_top_set_nat )
= top_top_set_set_int ) ) ).
% surjI
thf(fact_474_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_475_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_476_surjE,axiom,
! [F: set_int > int,Y: int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
=> ~ ! [X2: set_int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_477_surjE,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_478_surjE,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_479_surjE,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_480_surjE,axiom,
! [F: nat > set_int,Y: set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_481_surjE,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_482_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_483_surjD,axiom,
! [F: set_int > int,Y: int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
=> ? [X2: set_int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_484_surjD,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_485_surjD,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_486_surjD,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_487_surjD,axiom,
! [F: nat > set_int,Y: set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_488_surjD,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_489_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_490_Cantors__paradox,axiom,
! [A: set_int] :
~ ? [F3: int > set_int] :
( ( image_int_set_int @ F3 @ A )
= ( pow_int @ A ) ) ).
% Cantors_paradox
thf(fact_491_injD,axiom,
! [F: nat > set_int,X: nat,Y: nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_492_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_493_injI,axiom,
! [F: nat > set_int] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( inj_on_nat_set_int @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_494_injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_495_inj__eq,axiom,
! [F: nat > set_int,X: nat,Y: nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_496_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_497_inj__def,axiom,
! [F: nat > set_int] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
= ( ! [X3: nat,Y5: nat] :
( ( ( F @ X3 )
= ( F @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ).
% inj_def
thf(fact_498_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X3: nat,Y5: nat] :
( ( ( F @ X3 )
= ( F @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ).
% inj_def
thf(fact_499_the__inv__into__f__f,axiom,
! [F: nat > set_int,A: set_nat,X: nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_in879827501747159143et_int @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_500_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_501_the__inv__into__f__eq,axiom,
! [F: nat > set_int,A: set_nat,X: nat,Y: set_int] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( member_nat @ X @ A )
=> ( ( the_in879827501747159143et_int @ A @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_502_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_503_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_504_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_505_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_506_inj__on__image__Pow,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( inj_on8788328216580801005et_int @ ( image_set_int_int @ F ) @ ( pow_set_int @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_507_inj__on__image__Pow,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_508_inj__on__image__Pow,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( inj_on6285404204842837485et_int @ ( image_int_set_int @ F ) @ ( pow_int @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_509_inj__on__image__Pow,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_510_inj__on__image__Pow,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ ( pow_int @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_511_inj__on__image__Pow,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( inj_on286883514200038801et_int @ ( image_nat_set_int @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_512_inj__on__image__Pow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_513_range__subsetD,axiom,
! [F: set_int > int,B3: set_int,I: set_int] :
( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ top_top_set_set_int ) @ B3 )
=> ( member_int @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_514_range__subsetD,axiom,
! [F: int > int,B3: set_int,I: int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ top_top_set_int ) @ B3 )
=> ( member_int @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_515_range__subsetD,axiom,
! [F: nat > int,B3: set_int,I: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ top_top_set_nat ) @ B3 )
=> ( member_int @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_516_range__subsetD,axiom,
! [F: int > set_int,B3: set_set_int,I: int] :
( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ top_top_set_int ) @ B3 )
=> ( member_set_int @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_517_range__subsetD,axiom,
! [F: nat > set_int,B3: set_set_int,I: nat] :
( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ top_top_set_nat ) @ B3 )
=> ( member_set_int @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_518_range__subsetD,axiom,
! [F: int > nat,B3: set_nat,I: int] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ top_top_set_int ) @ B3 )
=> ( member_nat @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_519_range__subsetD,axiom,
! [F: nat > nat,B3: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B3 )
=> ( member_nat @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_520_range__ex1__eq,axiom,
! [F: set_int > int,B: int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( member_int @ B @ ( image_set_int_int @ F @ top_top_set_set_int ) )
= ( ? [X3: set_int] :
( ( B
= ( F @ X3 ) )
& ! [Y5: set_int] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_521_range__ex1__eq,axiom,
! [F: int > set_int,B: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( member_set_int @ B @ ( image_int_set_int @ F @ top_top_set_int ) )
= ( ? [X3: int] :
( ( B
= ( F @ X3 ) )
& ! [Y5: int] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_522_range__ex1__eq,axiom,
! [F: int > int,B: int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( member_int @ B @ ( image_int_int @ F @ top_top_set_int ) )
= ( ? [X3: int] :
( ( B
= ( F @ X3 ) )
& ! [Y5: int] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_523_range__ex1__eq,axiom,
! [F: int > nat,B: nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) )
= ( ? [X3: int] :
( ( B
= ( F @ X3 ) )
& ! [Y5: int] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_524_range__ex1__eq,axiom,
! [F: nat > set_int,B: set_int] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( member_set_int @ B @ ( image_nat_set_int @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_525_range__ex1__eq,axiom,
! [F: nat > int,B: int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_526_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_527_inj__image__eq__iff,axiom,
! [F: set_int > int,A: set_set_int,B3: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( ( image_set_int_int @ F @ A )
= ( image_set_int_int @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_528_inj__image__eq__iff,axiom,
! [F: int > set_int,A: set_int,B3: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ( image_int_set_int @ F @ A )
= ( image_int_set_int @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_529_inj__image__eq__iff,axiom,
! [F: int > nat,A: set_int,B3: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ( image_int_nat @ F @ A )
= ( image_int_nat @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_530_inj__image__eq__iff,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ( image_int_int @ F @ A )
= ( image_int_int @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_531_inj__image__eq__iff,axiom,
! [F: nat > int,A: set_nat,B3: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ( image_nat_int @ F @ A )
= ( image_nat_int @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_532_inj__image__eq__iff,axiom,
! [F: nat > set_int,A: set_nat,B3: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ( image_nat_set_int @ F @ A )
= ( image_nat_set_int @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_533_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B3 ) )
= ( A = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_534_inj__image__mem__iff,axiom,
! [F: set_int > set_int,A3: set_int,A: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_524474410958335435et_int @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_535_inj__image__mem__iff,axiom,
! [F: set_int > int,A3: set_int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( member_int @ ( F @ A3 ) @ ( image_set_int_int @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_536_inj__image__mem__iff,axiom,
! [F: set_int > nat,A3: set_int,A: set_set_int] :
( ( inj_on_set_int_nat @ F @ top_top_set_set_int )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_set_int_nat @ F @ A ) )
= ( member_set_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_537_inj__image__mem__iff,axiom,
! [F: int > set_int,A3: int,A: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_int_set_int @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_538_inj__image__mem__iff,axiom,
! [F: int > int,A3: int,A: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( member_int @ ( F @ A3 ) @ ( image_int_int @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_539_inj__image__mem__iff,axiom,
! [F: int > nat,A3: int,A: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_int_nat @ F @ A ) )
= ( member_int @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_540_inj__image__mem__iff,axiom,
! [F: nat > set_int,A3: nat,A: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( member_set_int @ ( F @ A3 ) @ ( image_nat_set_int @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_541_inj__image__mem__iff,axiom,
! [F: nat > int,A3: nat,A: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ ( F @ A3 ) @ ( image_nat_int @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_542_inj__image__mem__iff,axiom,
! [F: nat > nat,A3: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A3 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_543_Pow__mono,axiom,
! [A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( pow_int @ A ) @ ( pow_int @ B3 ) ) ) ).
% Pow_mono
thf(fact_544_Pow__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_le4317611570275147438et_int @ ( pow_set_int @ A ) @ ( pow_set_int @ B3 ) ) ) ).
% Pow_mono
thf(fact_545_Pow__mono,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B3 ) ) ) ).
% Pow_mono
thf(fact_546_inj__of__nat,axiom,
inj_on_nat_nat @ semiri1316708129612266289at_nat @ top_top_set_nat ).
% inj_of_nat
thf(fact_547_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_548_f__the__inv__into__f,axiom,
! [F: int > set_int,A: set_int,Y: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( member_set_int @ Y @ ( image_int_set_int @ F @ A ) )
=> ( ( F @ ( the_in1251481061984003779et_int @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_549_f__the__inv__into__f,axiom,
! [F: nat > set_int,A: set_nat,Y: set_int] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( member_set_int @ Y @ ( image_nat_set_int @ F @ A ) )
=> ( ( F @ ( the_in879827501747159143et_int @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_550_f__the__inv__into__f,axiom,
! [F: set_int > int,A: set_set_int,Y: int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( member_int @ Y @ ( image_set_int_int @ F @ A ) )
=> ( ( F @ ( the_in3676474720784958147nt_int @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_551_f__the__inv__into__f,axiom,
! [F: nat > int,A: set_nat,Y: int] :
( ( inj_on_nat_int @ F @ A )
=> ( ( member_int @ Y @ ( image_nat_int @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_int @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_552_f__the__inv__into__f,axiom,
! [F: int > int,A: set_int,Y: int] :
( ( inj_on_int_int @ F @ A )
=> ( ( member_int @ Y @ ( image_int_int @ F @ A ) )
=> ( ( F @ ( the_inv_into_int_int @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_553_f__the__inv__into__f,axiom,
! [F: int > nat,A: set_int,Y: nat] :
( ( inj_on_int_nat @ F @ A )
=> ( ( member_nat @ Y @ ( image_int_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_int_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_554_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_555_inj__on__the__inv__into,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( inj_on_int_set_int @ ( the_in3676474720784958147nt_int @ A @ F ) @ ( image_set_int_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_556_inj__on__the__inv__into,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( inj_on_int_nat @ ( the_inv_into_nat_int @ A @ F ) @ ( image_nat_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_557_inj__on__the__inv__into,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( inj_on_set_int_int @ ( the_in1251481061984003779et_int @ A @ F ) @ ( image_int_set_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_558_inj__on__the__inv__into,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( inj_on_nat_int @ ( the_inv_into_int_nat @ A @ F ) @ ( image_int_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_559_inj__on__the__inv__into,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( inj_on_int_int @ ( the_inv_into_int_int @ A @ F ) @ ( image_int_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_560_inj__on__the__inv__into,axiom,
! [F: set_int > nat,A: set_set_int] :
( ( inj_on_set_int_nat @ F @ A )
=> ( inj_on_nat_set_int @ ( the_in3678965191294008423nt_nat @ A @ F ) @ ( image_set_int_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_561_inj__on__the__inv__into,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( inj_on_set_int_nat @ ( the_in879827501747159143et_int @ A @ F ) @ ( image_nat_set_int @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_562_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_563_int_Ole__refl,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_int @ X @ X ) ) ).
% int.le_refl
thf(fact_564_int_Ole__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% int.le_antisym
thf(fact_565_UNIV_I4_J,axiom,
! [P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
& ( P @ X3 ) ) )
= ( ? [X5: int] : ( P @ X5 ) ) ) ).
% UNIV(4)
thf(fact_566_UNIV_I4_J,axiom,
! [P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ top_top_set_nat )
& ( P @ X3 ) ) )
= ( ? [X5: nat] : ( P @ X5 ) ) ) ).
% UNIV(4)
thf(fact_567_UNIV_I3_J,axiom,
! [P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( P @ X3 ) ) )
= ( ! [X5: int] : ( P @ X5 ) ) ) ).
% UNIV(3)
thf(fact_568_UNIV_I3_J,axiom,
! [P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ top_top_set_nat )
=> ( P @ X3 ) ) )
= ( ! [X5: nat] : ( P @ X5 ) ) ) ).
% UNIV(3)
thf(fact_569_iso__tuple__UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_570_iso__tuple__UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_571_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_572_UNIV_I2_J,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% UNIV(2)
thf(fact_573_UNIV_I2_J,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ A @ top_top_set_set_int ) ).
% UNIV(2)
thf(fact_574_UNIV_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% UNIV(2)
thf(fact_575_inj__image__Compl__subset,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ord_less_eq_set_int @ ( image_set_int_int @ F @ ( uminus7346710233107665121et_int @ A ) ) @ ( uminus1532241313380277803et_int @ ( image_set_int_int @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_576_inj__image__Compl__subset,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ ( uminus1532241313380277803et_int @ A ) ) @ ( uminus1532241313380277803et_int @ ( image_int_int @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_577_inj__image__Compl__subset,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus1532241313380277803et_int @ ( image_nat_int @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_578_inj__image__Compl__subset,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ ( uminus1532241313380277803et_int @ A ) ) @ ( uminus7346710233107665121et_int @ ( image_int_set_int @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_579_inj__image__Compl__subset,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus7346710233107665121et_int @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_580_inj__image__Compl__subset,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( uminus1532241313380277803et_int @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_int_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_581_inj__image__Compl__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_582_inj__on__image,axiom,
! [F: set_int > int,A: set_set_set_int] :
( ( inj_on_set_int_int @ F @ ( comple7281953568134767595et_int @ A ) )
=> ( inj_on8788328216580801005et_int @ ( image_set_int_int @ F ) @ A ) ) ).
% inj_on_image
thf(fact_583_inj__on__image,axiom,
! [F: nat > int,A: set_set_nat] :
( ( inj_on_nat_int @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ A ) ) ).
% inj_on_image
thf(fact_584_inj__on__image,axiom,
! [F: int > set_int,A: set_set_int] :
( ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ A ) )
=> ( inj_on6285404204842837485et_int @ ( image_int_set_int @ F ) @ A ) ) ).
% inj_on_image
thf(fact_585_inj__on__image,axiom,
! [F: int > nat,A: set_set_int] :
( ( inj_on_int_nat @ F @ ( comple3221217463730067765et_int @ A ) )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_586_inj__on__image,axiom,
! [F: int > int,A: set_set_int] :
( ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ A ) )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ A ) ) ).
% inj_on_image
thf(fact_587_inj__on__image,axiom,
! [F: nat > set_int,A: set_set_nat] :
( ( inj_on_nat_set_int @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on286883514200038801et_int @ ( image_nat_set_int @ F ) @ A ) ) ).
% inj_on_image
thf(fact_588_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_589_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_590_UnionI,axiom,
! [X6: set_set_int,C: set_set_set_int,A: set_int] :
( ( member_set_set_int @ X6 @ C )
=> ( ( member_set_int @ A @ X6 )
=> ( member_set_int @ A @ ( comple7281953568134767595et_int @ C ) ) ) ) ).
% UnionI
thf(fact_591_UnionI,axiom,
! [X6: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X6 @ C )
=> ( ( member_nat @ A @ X6 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_592_UnionI,axiom,
! [X6: set_int,C: set_set_int,A: int] :
( ( member_set_int @ X6 @ C )
=> ( ( member_int @ A @ X6 )
=> ( member_int @ A @ ( comple3221217463730067765et_int @ C ) ) ) ) ).
% UnionI
thf(fact_593_Union__iff,axiom,
! [A: set_int,C: set_set_set_int] :
( ( member_set_int @ A @ ( comple7281953568134767595et_int @ C ) )
= ( ? [X3: set_set_int] :
( ( member_set_set_int @ X3 @ C )
& ( member_set_int @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_594_Union__iff,axiom,
! [A: int,C: set_set_int] :
( ( member_int @ A @ ( comple3221217463730067765et_int @ C ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ C )
& ( member_int @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_595_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_596_ComplI,axiom,
! [C2: set_int,A: set_set_int] :
( ~ ( member_set_int @ C2 @ A )
=> ( member_set_int @ C2 @ ( uminus7346710233107665121et_int @ A ) ) ) ).
% ComplI
thf(fact_597_ComplI,axiom,
! [C2: int,A: set_int] :
( ~ ( member_int @ C2 @ A )
=> ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% ComplI
thf(fact_598_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_599_Compl__iff,axiom,
! [C2: set_int,A: set_set_int] :
( ( member_set_int @ C2 @ ( uminus7346710233107665121et_int @ A ) )
= ( ~ ( member_set_int @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_600_Compl__iff,axiom,
! [C2: int,A: set_int] :
( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) )
= ( ~ ( member_int @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_601_Compl__iff,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_602_inj__uminus,axiom,
! [A: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A ) ).
% inj_uminus
thf(fact_603_UN__ball__bex__simps_I4_J,axiom,
! [B3: nat > set_int,A: set_nat,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_604_UN__ball__bex__simps_I4_J,axiom,
! [B3: int > set_int,A: set_int,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ? [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_605_UN__ball__bex__simps_I2_J,axiom,
! [B3: nat > set_int,A: set_nat,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_606_UN__ball__bex__simps_I2_J,axiom,
! [B3: int > set_int,A: set_int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_607_bex__UN,axiom,
! [B3: nat > set_int,A: set_nat,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_608_bex__UN,axiom,
! [B3: int > set_int,A: set_int,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ? [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_609_ball__UN,axiom,
! [B3: nat > set_int,A: set_nat,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_610_ball__UN,axiom,
! [B3: int > set_int,A: set_int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y5: int] :
( ( member_int @ Y5 @ ( B3 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_611_Compl__subset__Compl__iff,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ A ) @ ( uminus7346710233107665121et_int @ B3 ) )
= ( ord_le4403425263959731960et_int @ B3 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_612_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B3 ) )
= ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_613_Compl__anti__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ B3 ) @ ( uminus7346710233107665121et_int @ A ) ) ) ).
% Compl_anti_mono
thf(fact_614_Compl__anti__mono,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_615_surj__uminus,axiom,
( ( image_int_int @ uminus_uminus_int @ top_top_set_int )
= top_top_set_int ) ).
% surj_uminus
thf(fact_616_Sup__UNIV,axiom,
( ( comple3221217463730067765et_int @ top_top_set_set_int )
= top_top_set_int ) ).
% Sup_UNIV
thf(fact_617_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_618_verit__negate__coefficient_I3_J,axiom,
! [A3: int,B: int] :
( ( A3 = B )
=> ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_619_ComplD,axiom,
! [C2: set_int,A: set_set_int] :
( ( member_set_int @ C2 @ ( uminus7346710233107665121et_int @ A ) )
=> ~ ( member_set_int @ C2 @ A ) ) ).
% ComplD
thf(fact_620_ComplD,axiom,
! [C2: int,A: set_int] :
( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) )
=> ~ ( member_int @ C2 @ A ) ) ).
% ComplD
thf(fact_621_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C2 @ A ) ) ).
% ComplD
thf(fact_622_UnionE,axiom,
! [A: set_int,C: set_set_set_int] :
( ( member_set_int @ A @ ( comple7281953568134767595et_int @ C ) )
=> ~ ! [X7: set_set_int] :
( ( member_set_int @ A @ X7 )
=> ~ ( member_set_set_int @ X7 @ C ) ) ) ).
% UnionE
thf(fact_623_UnionE,axiom,
! [A: int,C: set_set_int] :
( ( member_int @ A @ ( comple3221217463730067765et_int @ C ) )
=> ~ ! [X7: set_int] :
( ( member_int @ A @ X7 )
=> ~ ( member_set_int @ X7 @ C ) ) ) ).
% UnionE
thf(fact_624_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C ) ) ) ).
% UnionE
thf(fact_625_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_626_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_627_Sup__upper2,axiom,
! [U2: set_int,A: set_set_int,V: set_int] :
( ( member_set_int @ U2 @ A )
=> ( ( ord_less_eq_set_int @ V @ U2 )
=> ( ord_less_eq_set_int @ V @ ( comple3221217463730067765et_int @ A ) ) ) ) ).
% Sup_upper2
thf(fact_628_Sup__upper2,axiom,
! [U2: set_set_int,A: set_set_set_int,V: set_set_int] :
( ( member_set_set_int @ U2 @ A )
=> ( ( ord_le4403425263959731960et_int @ V @ U2 )
=> ( ord_le4403425263959731960et_int @ V @ ( comple7281953568134767595et_int @ A ) ) ) ) ).
% Sup_upper2
thf(fact_629_Sup__upper2,axiom,
! [U2: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U2 @ A )
=> ( ( ord_less_eq_set_nat @ V @ U2 )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_630_Sup__le__iff,axiom,
! [A: set_set_set_int,B: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ B )
= ( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A )
=> ( ord_le4403425263959731960et_int @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_631_Sup__le__iff,axiom,
! [A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_632_Sup__upper,axiom,
! [X: set_int,A: set_set_int] :
( ( member_set_int @ X @ A )
=> ( ord_less_eq_set_int @ X @ ( comple3221217463730067765et_int @ A ) ) ) ).
% Sup_upper
thf(fact_633_Sup__upper,axiom,
! [X: set_set_int,A: set_set_set_int] :
( ( member_set_set_int @ X @ A )
=> ( ord_le4403425263959731960et_int @ X @ ( comple7281953568134767595et_int @ A ) ) ) ).
% Sup_upper
thf(fact_634_Sup__upper,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_635_Sup__least,axiom,
! [A: set_set_int,Z2: set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( ord_less_eq_set_int @ X2 @ Z2 ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_636_Sup__least,axiom,
! [A: set_set_set_int,Z2: set_set_int] :
( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
=> ( ord_le4403425263959731960et_int @ X2 @ Z2 ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_637_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_638_Sup__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ! [A5: set_int] :
( ( member_set_int @ A5 @ A )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ B3 )
& ( ord_less_eq_set_int @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_639_Sup__mono,axiom,
! [A: set_set_set_int,B3: set_set_set_int] :
( ! [A5: set_set_int] :
( ( member_set_set_int @ A5 @ A )
=> ? [X4: set_set_int] :
( ( member_set_set_int @ X4 @ B3 )
& ( ord_le4403425263959731960et_int @ A5 @ X4 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_640_Sup__mono,axiom,
! [A: set_set_nat,B3: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Sup_mono
thf(fact_641_Sup__eqI,axiom,
! [A: set_set_int,X: set_int] :
( ! [Y3: set_int] :
( ( member_set_int @ Y3 @ A )
=> ( ord_less_eq_set_int @ Y3 @ X ) )
=> ( ! [Y3: set_int] :
( ! [Z3: set_int] :
( ( member_set_int @ Z3 @ A )
=> ( ord_less_eq_set_int @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_int @ X @ Y3 ) )
=> ( ( comple3221217463730067765et_int @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_642_Sup__eqI,axiom,
! [A: set_set_set_int,X: set_set_int] :
( ! [Y3: set_set_int] :
( ( member_set_set_int @ Y3 @ A )
=> ( ord_le4403425263959731960et_int @ Y3 @ X ) )
=> ( ! [Y3: set_set_int] :
( ! [Z3: set_set_int] :
( ( member_set_set_int @ Z3 @ A )
=> ( ord_le4403425263959731960et_int @ Z3 @ Y3 ) )
=> ( ord_le4403425263959731960et_int @ X @ Y3 ) )
=> ( ( comple7281953568134767595et_int @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_643_Sup__eqI,axiom,
! [A: set_set_nat,X: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [Y3: set_nat] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ A )
=> ( ord_less_eq_set_nat @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_644_SUP__cong,axiom,
! [A: set_set_int,B3: set_set_int,C: set_int > int,D: set_int > int] :
( ( A = B3 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_set_int_int @ C @ A ) )
= ( complete_Sup_Sup_int @ ( image_set_int_int @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_645_SUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > set_int,D: int > set_int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ C @ A ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_646_SUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > int,D: int > int] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ C @ A ) )
= ( complete_Sup_Sup_int @ ( image_int_int @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_647_SUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > set_int,D: nat > set_int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C @ A ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_648_SUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > int,D: nat > int] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ C @ A ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_649_SUP__cong,axiom,
! [A: set_set_int,B3: set_set_int,C: set_int > nat,D: set_int > nat] :
( ( A = B3 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_int_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_int_nat @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_650_SUP__cong,axiom,
! [A: set_int,B3: set_int,C: int > nat,D: int > nat] :
( ( A = B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_int_nat @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_651_SUP__cong,axiom,
! [A: set_nat,B3: set_nat,C: nat > nat,D: nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_652_Union__UNIV,axiom,
( ( comple3221217463730067765et_int @ top_top_set_set_int )
= top_top_set_int ) ).
% Union_UNIV
thf(fact_653_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_654_Union__subsetI,axiom,
! [A: set_set_int,B3: set_set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ? [Y4: set_int] :
( ( member_set_int @ Y4 @ B3 )
& ( ord_less_eq_set_int @ X2 @ Y4 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_655_Union__subsetI,axiom,
! [A: set_set_set_int,B3: set_set_set_int] :
( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
=> ? [Y4: set_set_int] :
( ( member_set_set_int @ Y4 @ B3 )
& ( ord_le4403425263959731960et_int @ X2 @ Y4 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_656_Union__subsetI,axiom,
! [A: set_set_nat,B3: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B3 )
& ( ord_less_eq_set_nat @ X2 @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Union_subsetI
thf(fact_657_Union__upper,axiom,
! [B3: set_int,A: set_set_int] :
( ( member_set_int @ B3 @ A )
=> ( ord_less_eq_set_int @ B3 @ ( comple3221217463730067765et_int @ A ) ) ) ).
% Union_upper
thf(fact_658_Union__upper,axiom,
! [B3: set_set_int,A: set_set_set_int] :
( ( member_set_set_int @ B3 @ A )
=> ( ord_le4403425263959731960et_int @ B3 @ ( comple7281953568134767595et_int @ A ) ) ) ).
% Union_upper
thf(fact_659_Union__upper,axiom,
! [B3: set_nat,A: set_set_nat] :
( ( member_set_nat @ B3 @ A )
=> ( ord_less_eq_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_660_Union__least,axiom,
! [A: set_set_int,C: set_int] :
( ! [X7: set_int] :
( ( member_set_int @ X7 @ A )
=> ( ord_less_eq_set_int @ X7 @ C ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ C ) ) ).
% Union_least
thf(fact_661_Union__least,axiom,
! [A: set_set_set_int,C: set_set_int] :
( ! [X7: set_set_int] :
( ( member_set_set_int @ X7 @ A )
=> ( ord_le4403425263959731960et_int @ X7 @ C ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ C ) ) ).
% Union_least
thf(fact_662_Union__least,axiom,
! [A: set_set_nat,C: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C ) ) ).
% Union_least
thf(fact_663_Sup__subset__mono,axiom,
! [A: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_664_Sup__subset__mono,axiom,
! [A: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_665_Sup__subset__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_666_SUP__eq,axiom,
! [A: set_int,B3: set_int,F: int > set_int,G: int > set_int] :
( ! [I2: int] :
( ( member_int @ I2 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B3 )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_667_SUP__eq,axiom,
! [A: set_int,B3: set_nat,F: int > set_int,G: nat > set_int] :
( ! [I2: int] :
( ( member_int @ I2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_668_SUP__eq,axiom,
! [A: set_nat,B3: set_int,F: nat > set_int,G: int > set_int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_669_SUP__eq,axiom,
! [A: set_nat,B3: set_nat,F: nat > set_int,G: nat > set_int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_670_SUP__eq,axiom,
! [A: set_int,B3: set_int,F: int > set_nat,G: int > set_nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B3 )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_671_SUP__eq,axiom,
! [A: set_int,B3: set_nat,F: int > set_nat,G: nat > set_nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_672_SUP__eq,axiom,
! [A: set_nat,B3: set_int,F: nat > set_nat,G: int > set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_673_SUP__eq,axiom,
! [A: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_674_SUP__eq,axiom,
! [A: set_set_int,B3: set_int,F: set_int > set_int,G: int > set_int] :
( ! [I2: set_int] :
( ( member_set_int @ I2 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B3 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_675_SUP__eq,axiom,
! [A: set_set_int,B3: set_nat,F: set_int > set_int,G: nat > set_int] :
( ! [I2: set_int] :
( ( member_set_int @ I2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_676_Union__mono,axiom,
! [A: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A @ B3 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_677_Union__mono,axiom,
! [A: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Union_mono
thf(fact_678_Union__mono,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_679_subset__Pow__Union,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ A @ ( pow_int @ ( comple3221217463730067765et_int @ A ) ) ) ).
% subset_Pow_Union
thf(fact_680_surj__Compl__image__subset,axiom,
! [F: set_int > int,A: set_set_int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_set_int_int @ F @ A ) ) @ ( image_set_int_int @ F @ ( uminus7346710233107665121et_int @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_681_surj__Compl__image__subset,axiom,
! [F: int > int,A: set_int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_int_int @ F @ A ) ) @ ( image_int_int @ F @ ( uminus1532241313380277803et_int @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_682_surj__Compl__image__subset,axiom,
! [F: nat > int,A: set_nat] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_nat_int @ F @ A ) ) @ ( image_nat_int @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_683_surj__Compl__image__subset,axiom,
! [F: int > set_int,A: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ ( image_int_set_int @ F @ A ) ) @ ( image_int_set_int @ F @ ( uminus1532241313380277803et_int @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_684_surj__Compl__image__subset,axiom,
! [F: nat > set_int,A: set_nat] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ ( image_nat_set_int @ F @ A ) ) @ ( image_nat_set_int @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_685_surj__Compl__image__subset,axiom,
! [F: int > nat,A: set_int] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_int_nat @ F @ A ) ) @ ( image_int_nat @ F @ ( uminus1532241313380277803et_int @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_686_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_687_int_Ototal__order__total,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% int.total_order_total
thf(fact_688_int_Ole__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ) ) ) ).
% int.le_trans
thf(fact_689_compl__le__compl__iff,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ X ) @ ( uminus7346710233107665121et_int @ Y ) )
= ( ord_le4403425263959731960et_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_690_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_691_neg__le__iff__le,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ B ) ) ).
% neg_le_iff_le
thf(fact_692_add_Oinverse__inverse,axiom,
! [A3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_693_neg__equal__iff__equal,axiom,
! [A3: int,B: int] :
( ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B ) )
= ( A3 = B ) ) ).
% neg_equal_iff_equal
thf(fact_694_int_Oadd_Oinv__closed,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( uminus_uminus_int @ X ) @ top_top_set_int ) ) ).
% int.add.inv_closed
thf(fact_695_int_Ominus__minus,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( uminus_uminus_int @ X ) )
= X ) ) ).
% int.minus_minus
thf(fact_696_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_697_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_698_equation__minus__iff,axiom,
! [A3: int,B: int] :
( ( A3
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_699_minus__equation__iff,axiom,
! [A3: int,B: int] :
( ( ( uminus_uminus_int @ A3 )
= B )
= ( ( uminus_uminus_int @ B )
= A3 ) ) ).
% minus_equation_iff
thf(fact_700_compl__le__swap2,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ Y ) @ X )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_701_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_702_compl__le__swap1,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ Y @ ( uminus7346710233107665121et_int @ X ) )
=> ( ord_le4403425263959731960et_int @ X @ ( uminus7346710233107665121et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_703_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_704_compl__mono,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ X @ Y )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ Y ) @ ( uminus7346710233107665121et_int @ X ) ) ) ).
% compl_mono
thf(fact_705_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_706_le__imp__neg__le,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_707_minus__le__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A3 ) ) ).
% minus_le_iff
thf(fact_708_le__minus__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A3 ) ) ) ).
% le_minus_iff
thf(fact_709_cSup__eq,axiom,
! [X6: set_int,A3: int] :
( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ( ord_less_eq_int @ X2 @ A3 ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ X4 @ Y3 ) )
=> ( ord_less_eq_int @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= A3 ) ) ) ).
% cSup_eq
thf(fact_710_cSup__eq__maximum,axiom,
! [Z2: set_int,X6: set_set_int] :
( ( member_set_int @ Z2 @ X6 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ X6 )
=> ( ord_less_eq_set_int @ X2 @ Z2 ) )
=> ( ( comple3221217463730067765et_int @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_711_cSup__eq__maximum,axiom,
! [Z2: set_set_int,X6: set_set_set_int] :
( ( member_set_set_int @ Z2 @ X6 )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ X6 )
=> ( ord_le4403425263959731960et_int @ X2 @ Z2 ) )
=> ( ( comple7281953568134767595et_int @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_712_cSup__eq__maximum,axiom,
! [Z2: int,X6: set_int] :
( ( member_int @ Z2 @ X6 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_713_cSup__eq__maximum,axiom,
! [Z2: set_nat,X6: set_set_nat] :
( ( member_set_nat @ Z2 @ X6 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X6 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_714_cSup__eq__maximum,axiom,
! [Z2: nat,X6: set_nat] :
( ( member_nat @ Z2 @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_715_top__empty__eq,axiom,
( top_top_set_int_o
= ( ^ [X3: set_int] : ( member_set_int @ X3 @ top_top_set_set_int ) ) ) ).
% top_empty_eq
thf(fact_716_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_717_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_718_Nats__def,axiom,
( semiring_1_Nats_nat
= ( image_nat_nat @ semiri1316708129612266289at_nat @ top_top_set_nat ) ) ).
% Nats_def
thf(fact_719_Nats__def,axiom,
( semiring_1_Nats_int
= ( image_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ) ) ).
% Nats_def
thf(fact_720_infinite__countable__subset,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ? [F4: nat > int] :
( ( inj_on_nat_int @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_int @ ( image_nat_int @ F4 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_721_infinite__countable__subset,axiom,
! [S: set_set_int] :
( ~ ( finite6197958912794628473et_int @ S )
=> ? [F4: nat > set_int] :
( ( inj_on_nat_set_int @ F4 @ top_top_set_nat )
& ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F4 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_722_infinite__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_countable_subset
thf(fact_723_infinite__iff__countable__subset,axiom,
! [S: set_int] :
( ( ~ ( finite_finite_int @ S ) )
= ( ? [F2: nat > int] :
( ( inj_on_nat_int @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_724_infinite__iff__countable__subset,axiom,
! [S: set_set_int] :
( ( ~ ( finite6197958912794628473et_int @ S ) )
= ( ? [F2: nat > set_int] :
( ( inj_on_nat_set_int @ F2 @ top_top_set_nat )
& ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_725_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_726_member__bind,axiom,
! [X: int,A: set_nat,F: nat > set_int] :
( ( member_int @ X @ ( bind_nat_int @ A @ F ) )
= ( member_int @ X @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% member_bind
thf(fact_727_member__bind,axiom,
! [X: int,A: set_int,F: int > set_int] :
( ( member_int @ X @ ( bind_int_int @ A @ F ) )
= ( member_int @ X @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A ) ) ) ) ).
% member_bind
thf(fact_728_image__uminus__greaterThan,axiom,
! [X: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or1207661135979820486an_int @ X ) )
= ( set_ord_lessThan_int @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_greaterThan
thf(fact_729_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_730_greaterThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_or1210151606488870762an_nat @ X )
= ( set_or1210151606488870762an_nat @ Y ) )
= ( X = Y ) ) ).
% greaterThan_eq_iff
thf(fact_731_finite__Plus__UNIV__iff,axiom,
( ( finite3009209376165618894nt_int @ top_to6358659424274202653nt_int )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_732_finite__Plus__UNIV__iff,axiom,
( ( finite7187060395674815602nt_nat @ top_to8848742569205929409nt_nat )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_733_finite__Plus__UNIV__iff,axiom,
( ( finite2009855664264564338at_int @ top_to4171737849581180865at_int )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_734_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_735_finite__imageI,axiom,
! [F5: set_set_int,H: set_int > int] :
( ( finite6197958912794628473et_int @ F5 )
=> ( finite_finite_int @ ( image_set_int_int @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_736_finite__imageI,axiom,
! [F5: set_nat,H: nat > set_int] :
( ( finite_finite_nat @ F5 )
=> ( finite6197958912794628473et_int @ ( image_nat_set_int @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_737_finite__imageI,axiom,
! [F5: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F5 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_738_finite__imageI,axiom,
! [F5: set_nat,H: nat > int] :
( ( finite_finite_nat @ F5 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_739_finite__imageI,axiom,
! [F5: set_int,H: int > set_int] :
( ( finite_finite_int @ F5 )
=> ( finite6197958912794628473et_int @ ( image_int_set_int @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_740_finite__imageI,axiom,
! [F5: set_int,H: int > nat] :
( ( finite_finite_int @ F5 )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_741_finite__imageI,axiom,
! [F5: set_int,H: int > int] :
( ( finite_finite_int @ F5 )
=> ( finite_finite_int @ ( image_int_int @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_742_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M4: set_nat] :
( ( member_set_nat @ M4 @ A )
=> ( finite_finite_nat @ M4 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_743_finite__Union,axiom,
! [A: set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ! [M4: set_int] :
( ( member_set_int @ M4 @ A )
=> ( finite_finite_int @ M4 ) )
=> ( finite_finite_int @ ( comple3221217463730067765et_int @ A ) ) ) ) ).
% finite_Union
thf(fact_744_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_745_finite__Pow__iff,axiom,
! [A: set_int] :
( ( finite6197958912794628473et_int @ ( pow_int @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_Pow_iff
thf(fact_746_greaterThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_or1207661135979820486an_int @ X ) @ ( set_or1207661135979820486an_int @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_747_greaterThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_748_finite__UN,axiom,
! [A: set_nat,B3: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite_finite_nat @ ( B3 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_749_finite__UN,axiom,
! [A: set_nat,B3: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite_finite_int @ ( B3 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_750_finite__UN,axiom,
! [A: set_int,B3: int > set_nat] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( finite_finite_nat @ ( B3 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_751_finite__UN,axiom,
! [A: set_int,B3: int > set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( finite_finite_int @ ( B3 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_752_finite__compl,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ ( uminus1532241313380277803et_int @ A ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_compl
thf(fact_753_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_754_image__uminus__lessThan,axiom,
! [X: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_ord_lessThan_int @ X ) )
= ( set_or1207661135979820486an_int @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_lessThan
thf(fact_755_finite__Prod__UNIV,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_756_finite__Prod__UNIV,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_757_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite1999359929226648326at_int @ top_to2179722763343057421at_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_758_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_759_finite__prod,axiom,
( ( finite2998713641127702882nt_int @ top_to4366644338036079209nt_int )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_760_finite__prod,axiom,
( ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_761_finite__prod,axiom,
( ( finite1999359929226648326at_int @ top_to2179722763343057421at_int )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_762_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_763_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_764_finite__UnionD,axiom,
! [A: set_set_int] :
( ( finite_finite_int @ ( comple3221217463730067765et_int @ A ) )
=> ( finite6197958912794628473et_int @ A ) ) ).
% finite_UnionD
thf(fact_765_Nats__infinite,axiom,
~ ( finite_finite_nat @ semiring_1_Nats_nat ) ).
% Nats_infinite
thf(fact_766_Nats__infinite,axiom,
~ ( finite_finite_int @ semiring_1_Nats_int ) ).
% Nats_infinite
thf(fact_767_infinite__Ioi,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_or1207661135979820486an_int @ A3 ) ) ).
% infinite_Ioi
thf(fact_768_infinite__Ioi,axiom,
! [A3: nat] :
~ ( finite_finite_nat @ ( set_or1210151606488870762an_nat @ A3 ) ) ).
% infinite_Ioi
thf(fact_769_finite__bind,axiom,
! [S: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( finite_finite_nat @ ( F @ X2 ) ) )
=> ( finite_finite_nat @ ( bind_nat_nat @ S @ F ) ) ) ) ).
% finite_bind
thf(fact_770_finite__bind,axiom,
! [S: set_nat,F: nat > set_int] :
( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( finite_finite_int @ ( F @ X2 ) ) )
=> ( finite_finite_int @ ( bind_nat_int @ S @ F ) ) ) ) ).
% finite_bind
thf(fact_771_finite__bind,axiom,
! [S: set_int,F: int > set_nat] :
( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( finite_finite_nat @ ( F @ X2 ) ) )
=> ( finite_finite_nat @ ( bind_int_nat @ S @ F ) ) ) ) ).
% finite_bind
thf(fact_772_finite__bind,axiom,
! [S: set_int,F: int > set_int] :
( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( finite_finite_int @ ( F @ X2 ) ) )
=> ( finite_finite_int @ ( bind_int_int @ S @ F ) ) ) ) ).
% finite_bind
thf(fact_773_Finite__Set_Ofinite__set,axiom,
( ( finite6197958912794628473et_int @ top_top_set_set_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% Finite_Set.finite_set
thf(fact_774_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_775_finite__has__minimal2,axiom,
! [A: set_set_int,A3: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( member_set_int @ A3 @ A )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( ord_less_eq_set_int @ X2 @ A3 )
& ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ A )
=> ( ( ord_less_eq_set_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_776_finite__has__minimal2,axiom,
! [A: set_set_set_int,A3: set_set_int] :
( ( finite4249678464180374575et_int @ A )
=> ( ( member_set_set_int @ A3 @ A )
=> ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
& ( ord_le4403425263959731960et_int @ X2 @ A3 )
& ! [Xa2: set_set_int] :
( ( member_set_set_int @ Xa2 @ A )
=> ( ( ord_le4403425263959731960et_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_777_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_778_finite__has__minimal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A3 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A )
=> ( ( ord_less_eq_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_779_finite__has__minimal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A3 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_780_finite__has__maximal2,axiom,
! [A: set_set_int,A3: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( member_set_int @ A3 @ A )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( ord_less_eq_set_int @ A3 @ X2 )
& ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ A )
=> ( ( ord_less_eq_set_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_781_finite__has__maximal2,axiom,
! [A: set_set_set_int,A3: set_set_int] :
( ( finite4249678464180374575et_int @ A )
=> ( ( member_set_set_int @ A3 @ A )
=> ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
& ( ord_le4403425263959731960et_int @ A3 @ X2 )
& ! [Xa2: set_set_int] :
( ( member_set_set_int @ Xa2 @ A )
=> ( ( ord_le4403425263959731960et_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_782_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_783_finite__has__maximal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A3 @ X2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_784_finite__has__maximal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A3 @ X2 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_785_ex__new__if__finite,axiom,
! [A: set_set_int] :
( ~ ( finite6197958912794628473et_int @ top_top_set_set_int )
=> ( ( finite6197958912794628473et_int @ A )
=> ? [A5: set_int] :
~ ( member_set_int @ A5 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_786_ex__new__if__finite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_int @ A )
=> ? [A5: int] :
~ ( member_int @ A5 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_787_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A5: nat] :
~ ( member_nat @ A5 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_788_infinite__UNIV__char__0,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_char_0
thf(fact_789_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_790_rev__finite__subset,axiom,
! [B3: set_int,A: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_791_rev__finite__subset,axiom,
! [B3: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( finite6197958912794628473et_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_792_rev__finite__subset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_793_infinite__super,axiom,
! [S: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S @ T2 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_794_infinite__super,axiom,
! [S: set_set_int,T2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ T2 )
=> ( ~ ( finite6197958912794628473et_int @ S )
=> ~ ( finite6197958912794628473et_int @ T2 ) ) ) ).
% infinite_super
thf(fact_795_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_796_finite__subset,axiom,
! [A: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_797_finite__subset,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( finite6197958912794628473et_int @ A ) ) ) ).
% finite_subset
thf(fact_798_finite__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_799_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_800_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_801_infinite__Iio,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A3 ) ) ).
% infinite_Iio
thf(fact_802_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_803_finite__subset__Union,axiom,
! [A: set_int,B6: set_set_int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ A @ ( comple3221217463730067765et_int @ B6 ) )
=> ~ ! [F6: set_set_int] :
( ( finite6197958912794628473et_int @ F6 )
=> ( ( ord_le4403425263959731960et_int @ F6 @ B6 )
=> ~ ( ord_less_eq_set_int @ A @ ( comple3221217463730067765et_int @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_804_finite__subset__Union,axiom,
! [A: set_set_int,B6: set_set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ord_le4403425263959731960et_int @ A @ ( comple7281953568134767595et_int @ B6 ) )
=> ~ ! [F6: set_set_set_int] :
( ( finite4249678464180374575et_int @ F6 )
=> ( ( ord_le4317611570275147438et_int @ F6 @ B6 )
=> ~ ( ord_le4403425263959731960et_int @ A @ ( comple7281953568134767595et_int @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_805_finite__subset__Union,axiom,
! [A: set_nat,B6: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B6 ) )
=> ~ ! [F6: set_set_nat] :
( ( finite1152437895449049373et_nat @ F6 )
=> ( ( ord_le6893508408891458716et_nat @ F6 @ B6 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_806_Nats__cases,axiom,
! [X: nat] :
( ( member_nat @ X @ semiring_1_Nats_nat )
=> ~ ! [N3: nat] :
( X
!= ( semiri1316708129612266289at_nat @ N3 ) ) ) ).
% Nats_cases
thf(fact_807_Nats__cases,axiom,
! [X: int] :
( ( member_int @ X @ semiring_1_Nats_int )
=> ~ ! [N3: nat] :
( X
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% Nats_cases
thf(fact_808_Nats__induct,axiom,
! [X: nat,P: nat > $o] :
( ( member_nat @ X @ semiring_1_Nats_nat )
=> ( ! [N3: nat] : ( P @ ( semiri1316708129612266289at_nat @ N3 ) )
=> ( P @ X ) ) ) ).
% Nats_induct
thf(fact_809_Nats__induct,axiom,
! [X: int,P: int > $o] :
( ( member_int @ X @ semiring_1_Nats_int )
=> ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ X ) ) ) ).
% Nats_induct
thf(fact_810_of__nat__in__Nats,axiom,
! [N: nat] : ( member_nat @ ( semiri1316708129612266289at_nat @ N ) @ semiring_1_Nats_nat ) ).
% of_nat_in_Nats
thf(fact_811_of__nat__in__Nats,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ semiring_1_Nats_int ) ).
% of_nat_in_Nats
thf(fact_812_finite__surj,axiom,
! [A: set_set_int,B3: set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_set_int_int @ F @ A ) )
=> ( finite_finite_int @ B3 ) ) ) ).
% finite_surj
thf(fact_813_finite__surj,axiom,
! [A: set_nat,B3: set_int,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ( finite_finite_int @ B3 ) ) ) ).
% finite_surj
thf(fact_814_finite__surj,axiom,
! [A: set_int,B3: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A ) )
=> ( finite_finite_int @ B3 ) ) ) ).
% finite_surj
thf(fact_815_finite__surj,axiom,
! [A: set_nat,B3: set_set_int,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_nat_set_int @ F @ A ) )
=> ( finite6197958912794628473et_int @ B3 ) ) ) ).
% finite_surj
thf(fact_816_finite__surj,axiom,
! [A: set_int,B3: set_set_int,F: int > set_int] :
( ( finite_finite_int @ A )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_int_set_int @ F @ A ) )
=> ( finite6197958912794628473et_int @ B3 ) ) ) ).
% finite_surj
thf(fact_817_finite__surj,axiom,
! [A: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_818_finite__surj,axiom,
! [A: set_int,B3: set_nat,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_819_finite__subset__image,axiom,
! [B3: set_int,F: int > int,A: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A ) )
=> ? [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
& ( finite_finite_int @ C3 )
& ( B3
= ( image_int_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_820_finite__subset__image,axiom,
! [B3: set_int,F: set_int > int,A: set_set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_set_int_int @ F @ A ) )
=> ? [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
& ( finite6197958912794628473et_int @ C3 )
& ( B3
= ( image_set_int_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_821_finite__subset__image,axiom,
! [B3: set_int,F: nat > int,A: set_nat] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B3
= ( image_nat_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_822_finite__subset__image,axiom,
! [B3: set_set_int,F: int > set_int,A: set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_int_set_int @ F @ A ) )
=> ? [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
& ( finite_finite_int @ C3 )
& ( B3
= ( image_int_set_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_823_finite__subset__image,axiom,
! [B3: set_set_int,F: set_int > set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_524474410958335435et_int @ F @ A ) )
=> ? [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
& ( finite6197958912794628473et_int @ C3 )
& ( B3
= ( image_524474410958335435et_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_824_finite__subset__image,axiom,
! [B3: set_set_int,F: nat > set_int,A: set_nat] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_nat_set_int @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B3
= ( image_nat_set_int @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_825_finite__subset__image,axiom,
! [B3: set_nat,F: int > nat,A: set_int] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A ) )
=> ? [C3: set_int] :
( ( ord_less_eq_set_int @ C3 @ A )
& ( finite_finite_int @ C3 )
& ( B3
= ( image_int_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_826_finite__subset__image,axiom,
! [B3: set_nat,F: set_int > nat,A: set_set_int] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_set_int_nat @ F @ A ) )
=> ? [C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A )
& ( finite6197958912794628473et_int @ C3 )
& ( B3
= ( image_set_int_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_827_finite__subset__image,axiom,
! [B3: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B3
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_828_ex__finite__subset__image,axiom,
! [F: int > int,A: set_int,P: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_829_ex__finite__subset__image,axiom,
! [F: set_int > int,A: set_set_int,P: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_set_int_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A )
& ( P @ ( image_set_int_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_830_ex__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_831_ex__finite__subset__image,axiom,
! [F: int > set_int,A: set_int,P: set_set_int > $o] :
( ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_int_set_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P @ ( image_int_set_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_832_ex__finite__subset__image,axiom,
! [F: set_int > set_int,A: set_set_int,P: set_set_int > $o] :
( ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_524474410958335435et_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A )
& ( P @ ( image_524474410958335435et_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_833_ex__finite__subset__image,axiom,
! [F: nat > set_int,A: set_nat,P: set_set_int > $o] :
( ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_nat_set_int @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_set_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_834_ex__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_835_ex__finite__subset__image,axiom,
! [F: set_int > nat,A: set_set_int,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_set_int_nat @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A )
& ( P @ ( image_set_int_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_836_ex__finite__subset__image,axiom,
! [F: 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 @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_837_all__finite__subset__image,axiom,
! [F: int > int,A: set_int,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_838_all__finite__subset__image,axiom,
! [F: set_int > int,A: set_set_int,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_set_int_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A ) )
=> ( P @ ( image_set_int_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_839_all__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_840_all__finite__subset__image,axiom,
! [F: int > set_int,A: set_int,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_int_set_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P @ ( image_int_set_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_841_all__finite__subset__image,axiom,
! [F: set_int > set_int,A: set_set_int,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_524474410958335435et_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A ) )
=> ( P @ ( image_524474410958335435et_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_842_all__finite__subset__image,axiom,
! [F: nat > set_int,A: set_nat,P: set_set_int > $o] :
( ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ ( image_nat_set_int @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_set_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_843_all__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_844_all__finite__subset__image,axiom,
! [F: set_int > nat,A: set_set_int,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_set_int_nat @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_int] :
( ( ( finite6197958912794628473et_int @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A ) )
=> ( P @ ( image_set_int_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_845_all__finite__subset__image,axiom,
! [F: 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 @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_846_le__cSup__finite,axiom,
! [X6: set_set_int,X: set_int] :
( ( finite6197958912794628473et_int @ X6 )
=> ( ( member_set_int @ X @ X6 )
=> ( ord_less_eq_set_int @ X @ ( comple3221217463730067765et_int @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_847_le__cSup__finite,axiom,
! [X6: set_set_set_int,X: set_set_int] :
( ( finite4249678464180374575et_int @ X6 )
=> ( ( member_set_set_int @ X @ X6 )
=> ( ord_le4403425263959731960et_int @ X @ ( comple7281953568134767595et_int @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_848_le__cSup__finite,axiom,
! [X6: set_int,X: int] :
( ( finite_finite_int @ X6 )
=> ( ( member_int @ X @ X6 )
=> ( ord_less_eq_int @ X @ ( complete_Sup_Sup_int @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_849_le__cSup__finite,axiom,
! [X6: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ X6 )
=> ( ( member_set_nat @ X @ X6 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_850_le__cSup__finite,axiom,
! [X6: set_nat,X: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_851_finite__image__iff,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( finite6197958912794628473et_int @ ( image_nat_set_int @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_852_finite__image__iff,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( finite6197958912794628473et_int @ ( image_int_set_int @ F @ A ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_image_iff
thf(fact_853_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_854_finite__image__iff,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_image_iff
thf(fact_855_finite__image__iff,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( finite_finite_int @ ( image_set_int_int @ F @ A ) )
= ( finite6197958912794628473et_int @ A ) ) ) ).
% finite_image_iff
thf(fact_856_finite__image__iff,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_857_finite__image__iff,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_image_iff
thf(fact_858_finite__imageD,axiom,
! [F: nat > set_int,A: set_nat] :
( ( finite6197958912794628473et_int @ ( image_nat_set_int @ F @ A ) )
=> ( ( inj_on_nat_set_int @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_859_finite__imageD,axiom,
! [F: int > set_int,A: set_int] :
( ( finite6197958912794628473et_int @ ( image_int_set_int @ F @ A ) )
=> ( ( inj_on_int_set_int @ F @ A )
=> ( finite_finite_int @ A ) ) ) ).
% finite_imageD
thf(fact_860_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_861_finite__imageD,axiom,
! [F: int > nat,A: set_int] :
( ( finite_finite_nat @ ( image_int_nat @ F @ A ) )
=> ( ( inj_on_int_nat @ F @ A )
=> ( finite_finite_int @ A ) ) ) ).
% finite_imageD
thf(fact_862_finite__imageD,axiom,
! [F: set_int > int,A: set_set_int] :
( ( finite_finite_int @ ( image_set_int_int @ F @ A ) )
=> ( ( inj_on_set_int_int @ F @ A )
=> ( finite6197958912794628473et_int @ A ) ) ) ).
% finite_imageD
thf(fact_863_finite__imageD,axiom,
! [F: nat > int,A: set_nat] :
( ( finite_finite_int @ ( image_nat_int @ F @ A ) )
=> ( ( inj_on_nat_int @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_864_finite__imageD,axiom,
! [F: int > int,A: set_int] :
( ( finite_finite_int @ ( image_int_int @ F @ A ) )
=> ( ( inj_on_int_int @ F @ A )
=> ( finite_finite_int @ A ) ) ) ).
% finite_imageD
thf(fact_865_bind__UNION,axiom,
( bind_nat_int
= ( ^ [A4: set_nat,F2: nat > set_int] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_866_bind__UNION,axiom,
( bind_int_int
= ( ^ [A4: set_int,F2: int > set_int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_867_finite__UNIV__surj__inj,axiom,
! [F: int > int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( inj_on_int_int @ F @ top_top_set_int ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_868_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_869_finite__UNIV__inj__surj,axiom,
! [F: int > int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_870_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_871_finite__surj__inj,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ A @ ( image_int_int @ F @ A ) )
=> ( inj_on_int_int @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_872_finite__surj__inj,axiom,
! [A: set_set_int,F: set_int > set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ord_le4403425263959731960et_int @ A @ ( image_524474410958335435et_int @ F @ A ) )
=> ( inj_on6435365835345961783et_int @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_873_finite__surj__inj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_874_inj__on__finite,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite6197958912794628473et_int @ A ) ) ) ) ).
% inj_on_finite
thf(fact_875_inj__on__finite,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( inj_on_nat_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_876_inj__on__finite,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_int @ A ) ) ) ) ).
% inj_on_finite
thf(fact_877_inj__on__finite,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_878_inj__on__finite,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( finite_finite_int @ A ) ) ) ) ).
% inj_on_finite
thf(fact_879_inj__on__finite,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_880_inj__on__finite,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( inj_on_int_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_int @ A ) ) ) ) ).
% inj_on_finite
thf(fact_881_endo__inj__surj,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ A )
=> ( ( inj_on_int_int @ F @ A )
=> ( ( image_int_int @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_882_endo__inj__surj,axiom,
! [A: set_set_int,F: set_int > set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A ) @ A )
=> ( ( inj_on6435365835345961783et_int @ F @ A )
=> ( ( image_524474410958335435et_int @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_883_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_884_Sup__SUP__eq,axiom,
( comple1561774118418404442_int_o
= ( ^ [S2: set_set_int_o,X3: set_int] : ( member_set_int @ X3 @ ( comple7281953568134767595et_int @ ( image_9165537771349138752et_int @ collect_set_int @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_885_Sup__SUP__eq,axiom,
( comple6496622788309502864_int_o
= ( ^ [S2: set_int_o,X3: int] : ( member_int @ X3 @ ( comple3221217463730067765et_int @ ( image_int_o_set_int @ collect_int @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_886_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_887_finite__option__UNIV,axiom,
( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% finite_option_UNIV
thf(fact_888_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_889_range__inj__infinite,axiom,
! [F: nat > set_int] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ~ ( finite6197958912794628473et_int @ ( image_nat_set_int @ F @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_890_range__inj__infinite,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ~ ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_891_range__inj__infinite,axiom,
! [F: nat > int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ~ ( finite_finite_int @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_892_finite__field__axioms_Ointro,axiom,
! [R: partia4934656038542163276t_unit] :
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R ) )
=> ( ring_f1119117527023254578t_unit @ R ) ) ).
% finite_field_axioms.intro
thf(fact_893_finite__field__axioms__def,axiom,
( ring_f1119117527023254578t_unit
= ( ^ [R2: partia4934656038542163276t_unit] : ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% finite_field_axioms_def
thf(fact_894_inj__on__iff__card__le,axiom,
! [A: set_set_int,B3: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ? [F2: set_int > int] :
( ( inj_on_set_int_int @ F2 @ A )
& ( ord_less_eq_set_int @ ( image_set_int_int @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_895_inj__on__iff__card__le,axiom,
! [A: set_nat,B3: set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ? [F2: nat > int] :
( ( inj_on_nat_int @ F2 @ A )
& ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_896_inj__on__iff__card__le,axiom,
! [A: set_int,B3: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ? [F2: int > int] :
( ( inj_on_int_int @ F2 @ A )
& ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_897_inj__on__iff__card__le,axiom,
! [A: set_nat,B3: set_set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( ? [F2: nat > set_int] :
( ( inj_on_nat_set_int @ F2 @ A )
& ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_898_inj__on__iff__card__le,axiom,
! [A: set_int,B3: set_set_int] :
( ( finite_finite_int @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( ? [F2: int > set_int] :
( ( inj_on_int_set_int @ F2 @ A )
& ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_899_inj__on__iff__card__le,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_900_inj__on__iff__card__le,axiom,
! [A: set_int,B3: set_nat] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ? [F2: int > nat] :
( ( inj_on_int_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B3 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_901_card__inj__on__le,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_902_card__inj__on__le,axiom,
! [F: int > int,A: set_int,B3: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_903_card__inj__on__le,axiom,
! [F: nat > int,A: set_nat,B3: set_int] :
( ( inj_on_nat_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_904_card__inj__on__le,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_905_card__inj__on__le,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_906_card__inj__on__le,axiom,
! [F: int > nat,A: set_int,B3: set_nat] :
( ( inj_on_int_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_907_card__inj__on__le,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_908_card__le__inj,axiom,
! [A: set_set_int,B3: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_int @ B3 ) )
=> ? [F4: set_int > int] :
( ( ord_less_eq_set_int @ ( image_set_int_int @ F4 @ A ) @ B3 )
& ( inj_on_set_int_int @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_909_card__le__inj,axiom,
! [A: set_nat,B3: set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) )
=> ? [F4: nat > int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F4 @ A ) @ B3 )
& ( inj_on_nat_int @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_910_card__le__inj,axiom,
! [A: set_int,B3: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) )
=> ? [F4: int > int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F4 @ A ) @ B3 )
& ( inj_on_int_int @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_911_card__le__inj,axiom,
! [A: set_nat,B3: set_set_int] :
( ( finite_finite_nat @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_int @ B3 ) )
=> ? [F4: nat > set_int] :
( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F4 @ A ) @ B3 )
& ( inj_on_nat_set_int @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_912_card__le__inj,axiom,
! [A: set_int,B3: set_set_int] :
( ( finite_finite_int @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_set_int @ B3 ) )
=> ? [F4: int > set_int] :
( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F4 @ A ) @ B3 )
& ( inj_on_int_set_int @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_913_card__le__inj,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ? [F4: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ B3 )
& ( inj_on_nat_nat @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_914_card__le__inj,axiom,
! [A: set_int,B3: set_nat] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) )
=> ? [F4: int > nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F4 @ A ) @ B3 )
& ( inj_on_int_nat @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_915_card__lessThan,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U2 ) )
= U2 ) ).
% card_lessThan
thf(fact_916_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ top_top_set_int ) )
=> ( A = top_top_set_int ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_917_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_918_card__subset__eq,axiom,
! [B3: set_int,A: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_919_card__subset__eq,axiom,
! [B3: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( ( finite_card_set_int @ A )
= ( finite_card_set_int @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_920_card__subset__eq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_921_infinite__arbitrarily__large,axiom,
! [A: set_int,N: nat] :
( ~ ( finite_finite_int @ A )
=> ? [B7: set_int] :
( ( finite_finite_int @ B7 )
& ( ( finite_card_int @ B7 )
= N )
& ( ord_less_eq_set_int @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_922_infinite__arbitrarily__large,axiom,
! [A: set_set_int,N: nat] :
( ~ ( finite6197958912794628473et_int @ A )
=> ? [B7: set_set_int] :
( ( finite6197958912794628473et_int @ B7 )
& ( ( finite_card_set_int @ B7 )
= N )
& ( ord_le4403425263959731960et_int @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_923_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_924_card__le__if__inj__on__rel,axiom,
! [B3: set_set_int,A: set_set_int,R3: set_int > set_int > $o] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ! [A5: set_int] :
( ( member_set_int @ A5 @ A )
=> ? [B8: set_int] :
( ( member_set_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: set_int,A22: set_int,B5: set_int] :
( ( member_set_int @ A1 @ A )
=> ( ( member_set_int @ A22 @ A )
=> ( ( member_set_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_925_card__le__if__inj__on__rel,axiom,
! [B3: set_set_int,A: set_int,R3: int > set_int > $o] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ? [B8: set_int] :
( ( member_set_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B5: set_int] :
( ( member_int @ A1 @ A )
=> ( ( member_int @ A22 @ A )
=> ( ( member_set_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_926_card__le__if__inj__on__rel,axiom,
! [B3: set_set_int,A: set_nat,R3: nat > set_int > $o] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: set_int] :
( ( member_set_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_int] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_927_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_set_int,R3: set_int > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A5: set_int] :
( ( member_set_int @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: set_int,A22: set_int,B5: nat] :
( ( member_set_int @ A1 @ A )
=> ( ( member_set_int @ A22 @ A )
=> ( ( member_nat @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_928_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_int,R3: int > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B5: nat] :
( ( member_int @ A1 @ A )
=> ( ( member_int @ A22 @ A )
=> ( ( member_nat @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_929_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_930_card__le__if__inj__on__rel,axiom,
! [B3: set_int,A: set_set_int,R3: set_int > int > $o] :
( ( finite_finite_int @ B3 )
=> ( ! [A5: set_int] :
( ( member_set_int @ A5 @ A )
=> ? [B8: int] :
( ( member_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: set_int,A22: set_int,B5: int] :
( ( member_set_int @ A1 @ A )
=> ( ( member_set_int @ A22 @ A )
=> ( ( member_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_931_card__le__if__inj__on__rel,axiom,
! [B3: set_int,A: set_int,R3: int > int > $o] :
( ( finite_finite_int @ B3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A )
=> ? [B8: int] :
( ( member_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B5: int] :
( ( member_int @ A1 @ A )
=> ( ( member_int @ A22 @ A )
=> ( ( member_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_932_card__le__if__inj__on__rel,axiom,
! [B3: set_int,A: set_nat,R3: nat > int > $o] :
( ( finite_finite_int @ B3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: int] :
( ( member_int @ B8 @ B3 )
& ( R3 @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: int] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_int @ B5 @ B3 )
=> ( ( R3 @ A1 @ B5 )
=> ( ( R3 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_933_card__image,axiom,
! [F: set_int > int,A: set_set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( finite_card_int @ ( image_set_int_int @ F @ A ) )
= ( finite_card_set_int @ A ) ) ) ).
% card_image
thf(fact_934_card__image,axiom,
! [F: int > set_int,A: set_int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( finite_card_set_int @ ( image_int_set_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ).
% card_image
thf(fact_935_card__image,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ( finite_card_int @ ( image_int_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ).
% card_image
thf(fact_936_card__image,axiom,
! [F: nat > int,A: set_nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( finite_card_int @ ( image_nat_int @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_937_card__image,axiom,
! [F: nat > set_int,A: set_nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( finite_card_set_int @ ( image_nat_set_int @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_938_card__image,axiom,
! [F: int > nat,A: set_int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_int_nat @ F @ A ) )
= ( finite_card_int @ A ) ) ) ).
% card_image
thf(fact_939_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_940_finite__if__finite__subsets__card__bdd,axiom,
! [F5: set_int,C: nat] :
( ! [G2: set_int] :
( ( ord_less_eq_set_int @ G2 @ F5 )
=> ( ( finite_finite_int @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ G2 ) @ C ) ) )
=> ( ( finite_finite_int @ F5 )
& ( ord_less_eq_nat @ ( finite_card_int @ F5 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_941_finite__if__finite__subsets__card__bdd,axiom,
! [F5: set_set_int,C: nat] :
( ! [G2: set_set_int] :
( ( ord_le4403425263959731960et_int @ G2 @ F5 )
=> ( ( finite6197958912794628473et_int @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ G2 ) @ C ) ) )
=> ( ( finite6197958912794628473et_int @ F5 )
& ( ord_less_eq_nat @ ( finite_card_set_int @ F5 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_942_finite__if__finite__subsets__card__bdd,axiom,
! [F5: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F5 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F5 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F5 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_943_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
=> ~ ! [T3: set_int] :
( ( ord_less_eq_set_int @ T3 @ S )
=> ( ( ( finite_card_int @ T3 )
= N )
=> ~ ( finite_finite_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_944_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_int @ S ) )
=> ~ ! [T3: set_set_int] :
( ( ord_le4403425263959731960et_int @ T3 @ S )
=> ( ( ( finite_card_set_int @ T3 )
= N )
=> ~ ( finite6197958912794628473et_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_945_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_946_exists__subset__between,axiom,
! [A: set_int,N: nat,C: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C ) )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( finite_finite_int @ C )
=> ? [B7: set_int] :
( ( ord_less_eq_set_int @ A @ B7 )
& ( ord_less_eq_set_int @ B7 @ C )
& ( ( finite_card_int @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_947_exists__subset__between,axiom,
! [A: set_set_int,N: nat,C: set_set_int] :
( ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_int @ C ) )
=> ( ( ord_le4403425263959731960et_int @ A @ C )
=> ( ( finite6197958912794628473et_int @ C )
=> ? [B7: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B7 )
& ( ord_le4403425263959731960et_int @ B7 @ C )
& ( ( finite_card_set_int @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_948_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_949_card__seteq,axiom,
! [B3: set_int,A: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_int @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_950_card__seteq,axiom,
! [B3: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ B3 ) @ ( finite_card_set_int @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_951_card__seteq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_952_card__mono,axiom,
! [B3: set_int,A: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B3 ) ) ) ) ).
% card_mono
thf(fact_953_card__mono,axiom,
! [B3: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
=> ( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_set_int @ B3 ) ) ) ) ).
% card_mono
thf(fact_954_card__mono,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_955_card__image__le,axiom,
! [A: set_set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ A )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_set_int_int @ F @ A ) ) @ ( finite_card_set_int @ A ) ) ) ).
% card_image_le
thf(fact_956_card__image__le,axiom,
! [A: set_nat,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ ( image_nat_set_int @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_957_card__image__le,axiom,
! [A: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_958_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_959_card__image__le,axiom,
! [A: set_int,F: int > set_int] :
( ( finite_finite_int @ A )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ ( image_int_set_int @ F @ A ) ) @ ( finite_card_int @ A ) ) ) ).
% card_image_le
thf(fact_960_card__image__le,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_int @ F @ A ) ) @ ( finite_card_int @ A ) ) ) ).
% card_image_le
thf(fact_961_card__image__le,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_nat @ F @ A ) ) @ ( finite_card_int @ A ) ) ) ).
% card_image_le
thf(fact_962_eq__card__imp__inj__on,axiom,
! [A: set_set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ( finite_card_int @ ( image_set_int_int @ F @ A ) )
= ( finite_card_set_int @ A ) )
=> ( inj_on_set_int_int @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_963_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_int @ ( image_nat_int @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_int @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_964_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_set_int @ ( image_nat_set_int @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_set_int @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_965_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_966_eq__card__imp__inj__on,axiom,
! [A: set_int,F: int > set_int] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_set_int @ ( image_int_set_int @ F @ A ) )
= ( finite_card_int @ A ) )
=> ( inj_on_int_set_int @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_967_eq__card__imp__inj__on,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_int @ ( image_int_int @ F @ A ) )
= ( finite_card_int @ A ) )
=> ( inj_on_int_int @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_968_eq__card__imp__inj__on,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_nat @ ( image_int_nat @ F @ A ) )
= ( finite_card_int @ A ) )
=> ( inj_on_int_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_969_inj__on__iff__eq__card,axiom,
! [A: set_set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( inj_on_set_int_int @ F @ A )
= ( ( finite_card_int @ ( image_set_int_int @ F @ A ) )
= ( finite_card_set_int @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_970_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_int @ F @ A )
= ( ( finite_card_int @ ( image_nat_int @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_971_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_set_int @ F @ A )
= ( ( finite_card_set_int @ ( image_nat_set_int @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_972_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F @ A )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_973_inj__on__iff__eq__card,axiom,
! [A: set_int,F: int > set_int] :
( ( finite_finite_int @ A )
=> ( ( inj_on_int_set_int @ F @ A )
= ( ( finite_card_set_int @ ( image_int_set_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_974_inj__on__iff__eq__card,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( inj_on_int_int @ F @ A )
= ( ( finite_card_int @ ( image_int_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_975_inj__on__iff__eq__card,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( inj_on_int_nat @ F @ A )
= ( ( finite_card_nat @ ( image_int_nat @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_976_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_977_finite__nat__bounded,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_978_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S2: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_979_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_980_surj__card__le,axiom,
! [A: set_set_int,B3: set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_set_int_int @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_set_int @ A ) ) ) ) ).
% surj_card_le
thf(fact_981_surj__card__le,axiom,
! [A: set_nat,B3: set_int,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_982_surj__card__le,axiom,
! [A: set_int,B3: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_int @ A ) ) ) ) ).
% surj_card_le
thf(fact_983_surj__card__le,axiom,
! [A: set_nat,B3: set_set_int,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_nat_set_int @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_984_surj__card__le,axiom,
! [A: set_int,B3: set_set_int,F: int > set_int] :
( ( finite_finite_int @ A )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( image_int_set_int @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ B3 ) @ ( finite_card_int @ A ) ) ) ) ).
% surj_card_le
thf(fact_985_surj__card__le,axiom,
! [A: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_986_surj__card__le,axiom,
! [A: set_int,B3: set_nat,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_int @ A ) ) ) ) ).
% surj_card_le
thf(fact_987_card__bij__eq,axiom,
! [F: int > int,A: set_int,B3: set_int,G: int > int] :
( ( inj_on_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B3 )
=> ( ( inj_on_int_int @ G @ B3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ B3 ) @ A )
=> ( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( finite_card_int @ A )
= ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_988_card__bij__eq,axiom,
! [F: set_int > int,A: set_set_int,B3: set_int,G: int > set_int] :
( ( inj_on_set_int_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A ) @ B3 )
=> ( ( inj_on_int_set_int @ G @ B3 )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ G @ B3 ) @ A )
=> ( ( finite6197958912794628473et_int @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( finite_card_set_int @ A )
= ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_989_card__bij__eq,axiom,
! [F: nat > int,A: set_nat,B3: set_int,G: int > nat] :
( ( inj_on_nat_int @ F @ A )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B3 )
=> ( ( inj_on_int_nat @ G @ B3 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ B3 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( finite_card_nat @ A )
= ( finite_card_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_990_card__bij__eq,axiom,
! [F: int > set_int,A: set_int,B3: set_set_int,G: set_int > int] :
( ( inj_on_int_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A ) @ B3 )
=> ( ( inj_on_set_int_int @ G @ B3 )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ G @ B3 ) @ A )
=> ( ( finite_finite_int @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( finite_card_int @ A )
= ( finite_card_set_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_991_card__bij__eq,axiom,
! [F: set_int > set_int,A: set_set_int,B3: set_set_int,G: set_int > set_int] :
( ( inj_on6435365835345961783et_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A ) @ B3 )
=> ( ( inj_on6435365835345961783et_int @ G @ B3 )
=> ( ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ G @ B3 ) @ A )
=> ( ( finite6197958912794628473et_int @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( finite_card_set_int @ A )
= ( finite_card_set_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_992_card__bij__eq,axiom,
! [F: nat > set_int,A: set_nat,B3: set_set_int,G: set_int > nat] :
( ( inj_on_nat_set_int @ F @ A )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A ) @ B3 )
=> ( ( inj_on_set_int_nat @ G @ B3 )
=> ( ( ord_less_eq_set_nat @ ( image_set_int_nat @ G @ B3 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite6197958912794628473et_int @ B3 )
=> ( ( finite_card_nat @ A )
= ( finite_card_set_int @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_993_card__bij__eq,axiom,
! [F: int > nat,A: set_int,B3: set_nat,G: nat > int] :
( ( inj_on_int_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A ) @ B3 )
=> ( ( inj_on_nat_int @ G @ B3 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ B3 ) @ A )
=> ( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( finite_card_int @ A )
= ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_994_card__bij__eq,axiom,
! [F: set_int > nat,A: set_set_int,B3: set_nat,G: nat > set_int] :
( ( inj_on_set_int_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_set_int_nat @ F @ A ) @ B3 )
=> ( ( inj_on_nat_set_int @ G @ B3 )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ G @ B3 ) @ A )
=> ( ( finite6197958912794628473et_int @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( finite_card_set_int @ A )
= ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_995_card__bij__eq,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
=> ( ( inj_on_nat_nat @ G @ B3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B3 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( finite_card_nat @ A )
= ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_996_surjective__iff__injective__gen,axiom,
! [S: set_set_int,T2: set_int,F: set_int > int] :
( ( finite6197958912794628473et_int @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ( ( finite_card_set_int @ S )
= ( finite_card_int @ T2 ) )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ S ) @ T2 )
=> ( ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ? [Y5: set_int] :
( ( member_set_int @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_set_int_int @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_997_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_int,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_int @ T2 ) )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ S ) @ T2 )
=> ( ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_nat_int @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_998_surjective__iff__injective__gen,axiom,
! [S: set_int,T2: set_int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ( ( finite_card_int @ S )
= ( finite_card_int @ T2 ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ S ) @ T2 )
=> ( ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_int_int @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_999_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_set_int,F: nat > set_int] :
( ( finite_finite_nat @ S )
=> ( ( finite6197958912794628473et_int @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_set_int @ T2 ) )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ S ) @ T2 )
=> ( ( ! [X3: set_int] :
( ( member_set_int @ X3 @ T2 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_nat_set_int @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1000_surjective__iff__injective__gen,axiom,
! [S: set_int,T2: set_set_int,F: int > set_int] :
( ( finite_finite_int @ S )
=> ( ( finite6197958912794628473et_int @ T2 )
=> ( ( ( finite_card_int @ S )
= ( finite_card_set_int @ T2 ) )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ S ) @ T2 )
=> ( ( ! [X3: set_int] :
( ( member_set_int @ X3 @ T2 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_int_set_int @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1001_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S ) @ T2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_nat_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1002_surjective__iff__injective__gen,axiom,
! [S: set_int,T2: set_nat,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_int @ S )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ S ) @ T2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ S )
& ( ( F @ Y5 )
= X3 ) ) ) )
= ( inj_on_int_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1003_enumerate__mono__le__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1004_card__vimage__inj,axiom,
! [F: set_int > int,A: set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( ord_less_eq_set_int @ A @ ( image_set_int_int @ F @ top_top_set_set_int ) )
=> ( ( finite_card_set_int @ ( vimage_set_int_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1005_card__vimage__inj,axiom,
! [F: int > int,A: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ A @ ( image_int_int @ F @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage_int_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1006_card__vimage__inj,axiom,
! [F: nat > int,A: set_int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ A @ ( image_nat_int @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_int @ F @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1007_card__vimage__inj,axiom,
! [F: int > set_int,A: set_set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ord_le4403425263959731960et_int @ A @ ( image_int_set_int @ F @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage_int_set_int @ F @ A ) )
= ( finite_card_set_int @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1008_card__vimage__inj,axiom,
! [F: nat > set_int,A: set_set_int] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ord_le4403425263959731960et_int @ A @ ( image_nat_set_int @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_set_int @ F @ A ) )
= ( finite_card_set_int @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1009_card__vimage__inj,axiom,
! [F: int > nat,A: set_nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ A @ ( image_int_nat @ F @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage_int_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1010_card__vimage__inj,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1011_inj__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( inj_on_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat ) ) ).
% inj_enumerate
thf(fact_1012_Sup__greaterThanAtLeast,axiom,
! [X: set_int] :
( ( ord_less_set_int @ X @ top_top_set_int )
=> ( ( comple3221217463730067765et_int @ ( set_or5504389134266731388et_int @ X ) )
= top_top_set_int ) ) ).
% Sup_greaterThanAtLeast
thf(fact_1013_Sup__greaterThanAtLeast,axiom,
! [X: set_nat] :
( ( ord_less_set_nat @ X @ top_top_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( set_or458868116921152288et_nat @ X ) )
= top_top_set_nat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_1014_finite__field_Ofinite__carrier,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_f302724563095964181t_unit @ R )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R ) ) ) ).
% finite_field.finite_carrier
thf(fact_1015_psubsetI,axiom,
! [A: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_set_int @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_1016_psubsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_1017_vimageI,axiom,
! [F: set_int > set_int,A3: set_int,B: set_int,B3: set_set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_set_int @ B @ B3 )
=> ( member_set_int @ A3 @ ( vimage6596094510776989313et_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1018_vimageI,axiom,
! [F: int > set_int,A3: int,B: set_int,B3: set_set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_set_int @ B @ B3 )
=> ( member_int @ A3 @ ( vimage_int_set_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1019_vimageI,axiom,
! [F: nat > set_int,A3: nat,B: set_int,B3: set_set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_set_int @ B @ B3 )
=> ( member_nat @ A3 @ ( vimage_nat_set_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1020_vimageI,axiom,
! [F: set_int > int,A3: set_int,B: int,B3: set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_int @ B @ B3 )
=> ( member_set_int @ A3 @ ( vimage_set_int_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1021_vimageI,axiom,
! [F: int > int,A3: int,B: int,B3: set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_int @ B @ B3 )
=> ( member_int @ A3 @ ( vimage_int_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1022_vimageI,axiom,
! [F: nat > int,A3: nat,B: int,B3: set_int] :
( ( ( F @ A3 )
= B )
=> ( ( member_int @ B @ B3 )
=> ( member_nat @ A3 @ ( vimage_nat_int @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1023_vimageI,axiom,
! [F: set_int > nat,A3: set_int,B: nat,B3: set_nat] :
( ( ( F @ A3 )
= B )
=> ( ( member_nat @ B @ B3 )
=> ( member_set_int @ A3 @ ( vimage_set_int_nat @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1024_vimageI,axiom,
! [F: int > nat,A3: int,B: nat,B3: set_nat] :
( ( ( F @ A3 )
= B )
=> ( ( member_nat @ B @ B3 )
=> ( member_int @ A3 @ ( vimage_int_nat @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1025_vimageI,axiom,
! [F: nat > nat,A3: nat,B: nat,B3: set_nat] :
( ( ( F @ A3 )
= B )
=> ( ( member_nat @ B @ B3 )
=> ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_1026_vimage__eq,axiom,
! [A3: set_int,F: set_int > set_int,B3: set_set_int] :
( ( member_set_int @ A3 @ ( vimage6596094510776989313et_int @ F @ B3 ) )
= ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1027_vimage__eq,axiom,
! [A3: set_int,F: set_int > int,B3: set_int] :
( ( member_set_int @ A3 @ ( vimage_set_int_int @ F @ B3 ) )
= ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1028_vimage__eq,axiom,
! [A3: set_int,F: set_int > nat,B3: set_nat] :
( ( member_set_int @ A3 @ ( vimage_set_int_nat @ F @ B3 ) )
= ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1029_vimage__eq,axiom,
! [A3: int,F: int > set_int,B3: set_set_int] :
( ( member_int @ A3 @ ( vimage_int_set_int @ F @ B3 ) )
= ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1030_vimage__eq,axiom,
! [A3: int,F: int > int,B3: set_int] :
( ( member_int @ A3 @ ( vimage_int_int @ F @ B3 ) )
= ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1031_vimage__eq,axiom,
! [A3: int,F: int > nat,B3: set_nat] :
( ( member_int @ A3 @ ( vimage_int_nat @ F @ B3 ) )
= ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1032_vimage__eq,axiom,
! [A3: nat,F: nat > set_int,B3: set_set_int] :
( ( member_nat @ A3 @ ( vimage_nat_set_int @ F @ B3 ) )
= ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1033_vimage__eq,axiom,
! [A3: nat,F: nat > int,B3: set_int] :
( ( member_nat @ A3 @ ( vimage_nat_int @ F @ B3 ) )
= ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1034_vimage__eq,axiom,
! [A3: nat,F: nat > nat,B3: set_nat] :
( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B3 ) )
= ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimage_eq
thf(fact_1035_n__gt__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_gt_0
thf(fact_1036_neg__less__iff__less,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ B ) ) ).
% neg_less_iff_less
thf(fact_1037_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1038_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1039_lessThan__iff,axiom,
! [I: set_int,K: set_int] :
( ( member_set_int @ I @ ( set_or5935648273017318783et_int @ K ) )
= ( ord_less_set_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_1040_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_1041_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_1042_greaterThan__iff,axiom,
! [I: set_int,K: set_int] :
( ( member_set_int @ I @ ( set_or5504389134266731388et_int @ K ) )
= ( ord_less_set_int @ K @ I ) ) ).
% greaterThan_iff
thf(fact_1043_greaterThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_or1207661135979820486an_int @ K ) )
= ( ord_less_int @ K @ I ) ) ).
% greaterThan_iff
thf(fact_1044_greaterThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_1045_vimage__UNIV,axiom,
! [F: int > int] :
( ( vimage_int_int @ F @ top_top_set_int )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_1046_vimage__UNIV,axiom,
! [F: nat > int] :
( ( vimage_nat_int @ F @ top_top_set_int )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1047_vimage__UNIV,axiom,
! [F: int > nat] :
( ( vimage_int_nat @ F @ top_top_set_nat )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_1048_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1049_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1050_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1051_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1052_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1053_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1054_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1055_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1056_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1057_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_1058_vimageD,axiom,
! [A3: set_int,F: set_int > set_int,A: set_set_int] :
( ( member_set_int @ A3 @ ( vimage6596094510776989313et_int @ F @ A ) )
=> ( member_set_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1059_vimageD,axiom,
! [A3: set_int,F: set_int > int,A: set_int] :
( ( member_set_int @ A3 @ ( vimage_set_int_int @ F @ A ) )
=> ( member_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1060_vimageD,axiom,
! [A3: set_int,F: set_int > nat,A: set_nat] :
( ( member_set_int @ A3 @ ( vimage_set_int_nat @ F @ A ) )
=> ( member_nat @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1061_vimageD,axiom,
! [A3: int,F: int > set_int,A: set_set_int] :
( ( member_int @ A3 @ ( vimage_int_set_int @ F @ A ) )
=> ( member_set_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1062_vimageD,axiom,
! [A3: int,F: int > int,A: set_int] :
( ( member_int @ A3 @ ( vimage_int_int @ F @ A ) )
=> ( member_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1063_vimageD,axiom,
! [A3: int,F: int > nat,A: set_nat] :
( ( member_int @ A3 @ ( vimage_int_nat @ F @ A ) )
=> ( member_nat @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1064_vimageD,axiom,
! [A3: nat,F: nat > set_int,A: set_set_int] :
( ( member_nat @ A3 @ ( vimage_nat_set_int @ F @ A ) )
=> ( member_set_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1065_vimageD,axiom,
! [A3: nat,F: nat > int,A: set_int] :
( ( member_nat @ A3 @ ( vimage_nat_int @ F @ A ) )
=> ( member_int @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1066_vimageD,axiom,
! [A3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A3 ) @ A ) ) ).
% vimageD
thf(fact_1067_vimageE,axiom,
! [A3: set_int,F: set_int > set_int,B3: set_set_int] :
( ( member_set_int @ A3 @ ( vimage6596094510776989313et_int @ F @ B3 ) )
=> ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1068_vimageE,axiom,
! [A3: set_int,F: set_int > int,B3: set_int] :
( ( member_set_int @ A3 @ ( vimage_set_int_int @ F @ B3 ) )
=> ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1069_vimageE,axiom,
! [A3: set_int,F: set_int > nat,B3: set_nat] :
( ( member_set_int @ A3 @ ( vimage_set_int_nat @ F @ B3 ) )
=> ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1070_vimageE,axiom,
! [A3: int,F: int > set_int,B3: set_set_int] :
( ( member_int @ A3 @ ( vimage_int_set_int @ F @ B3 ) )
=> ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1071_vimageE,axiom,
! [A3: int,F: int > int,B3: set_int] :
( ( member_int @ A3 @ ( vimage_int_int @ F @ B3 ) )
=> ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1072_vimageE,axiom,
! [A3: int,F: int > nat,B3: set_nat] :
( ( member_int @ A3 @ ( vimage_int_nat @ F @ B3 ) )
=> ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1073_vimageE,axiom,
! [A3: nat,F: nat > set_int,B3: set_set_int] :
( ( member_nat @ A3 @ ( vimage_nat_set_int @ F @ B3 ) )
=> ( member_set_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1074_vimageE,axiom,
! [A3: nat,F: nat > int,B3: set_int] :
( ( member_nat @ A3 @ ( vimage_nat_int @ F @ B3 ) )
=> ( member_int @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1075_vimageE,axiom,
! [A3: nat,F: nat > nat,B3: set_nat] :
( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B3 ) )
=> ( member_nat @ ( F @ A3 ) @ B3 ) ) ).
% vimageE
thf(fact_1076_vimageI2,axiom,
! [F: set_int > nat,A3: set_int,A: set_nat] :
( ( member_nat @ ( F @ A3 ) @ A )
=> ( member_set_int @ A3 @ ( vimage_set_int_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1077_vimageI2,axiom,
! [F: int > nat,A3: int,A: set_nat] :
( ( member_nat @ ( F @ A3 ) @ A )
=> ( member_int @ A3 @ ( vimage_int_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1078_vimageI2,axiom,
! [F: nat > nat,A3: nat,A: set_nat] :
( ( member_nat @ ( F @ A3 ) @ A )
=> ( member_nat @ A3 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1079_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_1080_int_Olless__trans,axiom,
! [A3: int,B: int,C2: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ord_less_int @ A3 @ C2 ) ) ) ) ) ) ).
% int.lless_trans
thf(fact_1081_int_Olless__antisym,axiom,
! [A3: int,B: int] :
( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_int @ A3 @ B )
=> ~ ( ord_less_int @ B @ A3 ) ) ) ) ).
% int.lless_antisym
thf(fact_1082_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_1083_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_1084_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1085_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1086_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1087_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1088_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1089_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1090_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1091_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1092_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_1093_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_1094_Ring__Characteristic_Ozfact__iso__inj,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_nat_set_int @ ( ring_zfact_iso @ N ) @ ( set_ord_lessThan_nat @ N ) ) ) ).
% Ring_Characteristic.zfact_iso_inj
thf(fact_1095_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1096_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1097_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1098_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1099_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_1100_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1101_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1102_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1103_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1104_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1105_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1106_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1107_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1108_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1109_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_1110_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1111_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1112_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1113_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1114_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1115_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1116_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1117_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1118_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1119_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1120_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1121_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1122_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1123_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1124_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1125_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1126_int_Oadd_Oone__closed,axiom,
member_int @ zero_zero_int @ top_top_set_int ).
% int.add.one_closed
thf(fact_1127_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1128_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1129_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1130_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1131_int_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% int.minus_zero
thf(fact_1132_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1133_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1134_int_Oadd_Oinv__eq__1__iff,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( ( uminus_uminus_int @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ) ).
% int.add.inv_eq_1_iff
thf(fact_1135_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1136_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1137_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1138_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1139_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1140_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1141_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1142_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1143_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1144_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1145_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1146_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1147_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1148_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1149_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1150_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1151_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1152_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1153_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1154_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P2: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1155_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P2: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1156_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1157_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1158_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1159_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1160_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1161_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1162_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1163_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1164_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1165_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1166_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1167_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1168_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1169_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z4: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1170_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1171_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1172_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1173_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1174_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1175_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M3: nat] :
( M8
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1176_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1177_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1178_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1179_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1180_inj__Suc,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ suc @ N2 ) ).
% inj_Suc
thf(fact_1181_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1182_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1183_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1184_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1185_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1186_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1187_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1188_All__less__Suc2,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_less_Suc2
thf(fact_1189_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1190_Ex__less__Suc2,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_less_Suc2
thf(fact_1191_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1192_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1193_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1194_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1195_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1196_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1197_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1198_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1199_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1200_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1201_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1202_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1203_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1204_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1205_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1206_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1207_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_1208_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1209_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1210_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1211_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1212_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1213_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1214_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1215_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1216_infinite__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [R4: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R4 )
& ! [N6: nat] : ( member_nat @ ( R4 @ N6 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_1217_finite__enumerate,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [R4: nat > nat] :
( ( monotone_on_nat_nat @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) @ ord_less_nat @ ord_less_nat @ R4 )
& ! [N6: nat] :
( ( ord_less_nat @ N6 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( R4 @ N6 ) @ S ) ) ) ) ).
% finite_enumerate
thf(fact_1218_strict__mono__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).
% strict_mono_enumerate
thf(fact_1219_bij__betw__Suc,axiom,
! [M2: set_nat,N2: set_nat] :
( ( bij_betw_nat_nat @ suc @ M2 @ N2 )
= ( ( image_nat_nat @ suc @ M2 )
= N2 ) ) ).
% bij_betw_Suc
thf(fact_1220_bij__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).
% bij_enumerate
thf(fact_1221_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_1222_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1223_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1224_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1225_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1226_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1227_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1228_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1229_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1230_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1231_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1232_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1233_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1234_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1235_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1236_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1237_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1238_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1239_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1240_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1241_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1242_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1243_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1244_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1245_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1246_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1247_int_Om__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( times_times_int @ X @ Y ) @ top_top_set_int ) ) ) ).
% int.m_closed
thf(fact_1248_int_Or__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ X @ zero_zero_int )
= zero_zero_int ) ) ).
% int.r_null
thf(fact_1249_int_Ol__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ zero_zero_int @ X )
= zero_zero_int ) ) ).
% int.l_null
thf(fact_1250_int_Ointegral__iff,axiom,
! [A3: int,B: int] :
( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral_iff
thf(fact_1251_int_Om__rcancel,axiom,
! [A3: int,B: int,C2: int] :
( ( A3 != zero_zero_int )
=> ( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ( ( times_times_int @ B @ A3 )
= ( times_times_int @ C2 @ A3 ) )
= ( B = C2 ) ) ) ) ) ) ).
% int.m_rcancel
thf(fact_1252_int_Om__lcancel,axiom,
! [A3: int,B: int,C2: int] :
( ( A3 != zero_zero_int )
=> ( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ( ( times_times_int @ A3 @ B )
= ( times_times_int @ A3 @ C2 ) )
= ( B = C2 ) ) ) ) ) ) ).
% int.m_lcancel
thf(fact_1253_int_Ointegral,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
=> ( ( member_int @ A3 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral
thf(fact_1254_int_Ol__minus,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ ( uminus_uminus_int @ X ) @ Y )
= ( uminus_uminus_int @ ( times_times_int @ X @ Y ) ) ) ) ) ).
% int.l_minus
thf(fact_1255_int_Or__minus,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X @ ( uminus_uminus_int @ Y ) )
= ( uminus_uminus_int @ ( times_times_int @ X @ Y ) ) ) ) ) ).
% int.r_minus
thf(fact_1256_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1257_int_Om__lcomm,axiom,
! [X: int,Y: int,Z2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ( times_times_int @ X @ ( times_times_int @ Y @ Z2 ) )
= ( times_times_int @ Y @ ( times_times_int @ X @ Z2 ) ) ) ) ) ) ).
% int.m_lcomm
thf(fact_1258_int_Om__assoc,axiom,
! [X: int,Y: int,Z2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ( times_times_int @ ( times_times_int @ X @ Y ) @ Z2 )
= ( times_times_int @ X @ ( times_times_int @ Y @ Z2 ) ) ) ) ) ) ).
% int.m_assoc
thf(fact_1259_int_Om__comm,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) ) ) ) ).
% int.m_comm
thf(fact_1260_int__ops_I7_J,axiom,
! [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1261_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1262_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1263_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1264_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1265_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1266_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1267_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( image_nat_set_int @ ( ring_zfact_iso @ n ) @ ( set_ord_lessThan_nat @ n ) )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
%------------------------------------------------------------------------------