TPTP Problem File: SLH0232^1.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 : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00180_005897__12098646_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1398 ( 611 unt; 120 typ; 0 def)
% Number of atoms : 3640 (1348 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11972 ( 370 ~; 68 |; 347 &;9631 @)
% ( 0 <=>;1556 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 831 ( 831 >; 0 *; 0 +; 0 <<)
% Number of symbols : 115 ( 112 usr; 14 con; 0-5 aty)
% Number of variables : 3778 ( 323 ^;3309 !; 146 ?;3778 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:19:54.787
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (112)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
finite_finite_num: set_num > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
nO_MAT2475032472373502585et_nat: set_nat > set_nat > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
nO_MAT2968510229921005611_set_a: set_nat > set_a > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
nO_MAT4391312325019038905et_nat: set_a > set_nat > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
nO_MATCH_set_a_set_a: set_a > set_a > $o ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Num__Onum,type,
lattic2897619205827179199_a_num: ( a > num ) > set_a > a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
bot_bot_set_num: set_num ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Nat__Onat,type,
pluenn2073725187428264546up_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001t__Nat__Onat,type,
pluenn3669378163024332905et_nat: set_nat > ( nat > nat > nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001t__Nat__Onat,type,
pluenn5670965976768739049tp_nat: set_nat > ( nat > nat > nat ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OBex_001t__Nat__Onat,type,
bex_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__a,type,
bex_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_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_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_Aa____,type,
aa: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1270)
thf(fact_0_assms,axiom,
ord_less_eq_set_a @ a2 @ g ).
% assms
thf(fact_1__C0_Ohyps_C,axiom,
( zero_zero_nat
= ( finite_card_a @ aa ) ) ).
% "0.hyps"
thf(fact_2__C0_Oprems_C,axiom,
ord_less_eq_set_a @ aa @ g ).
% "0.prems"
thf(fact_3_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_4_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_5_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_6_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_7_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_8_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_9_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_10_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_11_associative,axiom,
! [A: a,B4: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C2 )
= ( addition @ A @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_12_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_13_card__sumset__0__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_14_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_15_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% sumset_def
thf(fact_16_sumsetp__sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_17_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_18_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_19_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_20_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_21_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_22_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_23_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_24_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_25_card__2__iff_H,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ S )
& ? [Y2: a] :
( ( member_a @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: a] :
( ( member_a @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_26_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z: nat] :
( ( member_nat @ Z @ S )
=> ( ( Z = X2 )
| ( Z = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_27_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_28_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_29_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_30_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_31_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_32_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_33_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_34_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_35_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_36_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_37_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_38_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_39_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_40_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_41_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_46_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_47_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_48_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_49_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_50_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_51_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_52_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_53_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_54_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_55_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_56_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_57_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_58_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_59_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_60_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_61_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_62_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_63_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_64_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_65_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_66_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_67_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_68_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_69_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_70_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_71_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_72_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_73_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_74_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_75_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_76_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_77_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_78_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_79_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_80_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_81_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_82_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M2: nat] :
( M5
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_83_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_84_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_85_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_86_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_87_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_88_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_89_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_90_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_91_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_92_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_93_card__le__sumset,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_94_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_95_card__sumset__0__iff_H,axiom,
! [A2: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_96_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_97_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_98_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_99_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_100_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_101_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_102_sumset__subset__Un2,axiom,
! [A2: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_103_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_104_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_105_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_106_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_107_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_108_finite__sumset_H,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset'
thf(fact_109_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_110_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_111_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_112_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_113_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_114_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_115_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_116_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_117_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_118_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_119_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_120_card__insert__disjoint,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_121_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_122_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_123_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_124_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_125_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_126_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_127_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_128_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_129_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_130_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_131_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_132_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_133_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_134_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_135_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_136_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_137_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_138_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_139_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_140_finite__has__maximal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ A @ X3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_141_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_142_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_143_finite__has__minimal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ X3 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_144_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_145_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_146_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_147_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_148_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_149_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_150_card__insert__if,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( finite_card_a @ A2 ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_151_card__insert__if,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( finite_card_nat @ A2 ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_152_card__Suc__eq__finite,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A2
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_153_card__Suc__eq__finite,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B3: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ B3 @ B6 ) )
& ~ ( member_nat @ B3 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_154_card__le__Suc__iff,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A2 ) )
= ( ? [A4: a,B6: set_a] :
( ( A2
= ( insert_a @ A4 @ B6 ) )
& ~ ( member_a @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B6 ) )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_155_card__le__Suc__iff,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A2 ) )
= ( ? [A4: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ A4 @ B6 ) )
& ~ ( member_nat @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_156_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_157_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_158_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_159_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_160_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_161_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N )
& ( ord_less_eq_set_a @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_162_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_163_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_164_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_165_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_166_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_167_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ B7 )
& ( ord_less_eq_set_a @ B7 @ C )
& ( ( finite_card_a @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_168_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_169_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_170_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_171_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_172_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B2: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_173_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B2: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_174_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B2: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_175_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B2: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B2 @ B )
=> ( ( R2 @ A1 @ B2 )
=> ( ( R2 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_176_card__le__Suc0__iff__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_177_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_178_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_179_n__subsets,axiom,
! [A2: set_nat,K: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_set_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
& ( ( finite_card_nat @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card_nat @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_180_n__subsets,axiom,
! [A2: set_a,K: nat] :
( ( finite_finite_a @ A2 )
=> ( ( finite_card_set_a
@ ( collect_set_a
@ ^ [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
& ( ( finite_card_a @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card_a @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_181_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_182_card__sumset__singleton__eq,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_183_card__sumset__le,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_184_infinite__sumset__aux,axiom,
! [A2: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_185_infinite__sumset__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_186_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_187_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_188_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_189_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_190_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_191_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_192_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_193_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_194_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_195_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_196_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_197_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_198_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_199_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_200_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_201_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_202_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_203_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_204_insertCI,axiom,
! [A: nat,B: set_nat,B4: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B4 ) )
=> ( member_nat @ A @ ( insert_nat @ B4 @ B ) ) ) ).
% insertCI
thf(fact_205_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_206_insert__iff,axiom,
! [A: nat,B4: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_207_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_208_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_209_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_210_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_211_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_212_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_213_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_214_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_215_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_216_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_217_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_218_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_219_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_220_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_221_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_222_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_223_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_224_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_225_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_226_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_227_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_228_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_229_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_230_Int__insert__left__if0,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_231_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_232_Int__insert__left__if1,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_233_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_234_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_235_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_236_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_237_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_238_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_239_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_240_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_241_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_242_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_243_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_244_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_245_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_246_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_247_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_248_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B4: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B4 @ bot_bot_set_nat ) )
= ( ( A = B4 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_249_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_250_singleton__insert__inj__eq,axiom,
! [B4: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B4 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_251_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_252_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_253_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B4: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B4 @ B ) ) )
= ( ~ ( member_nat @ B4 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_254_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_255_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_256_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_257_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_258_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_259_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_260_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_261_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_262_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_263_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_264_sumset__empty_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_265_sumset__empty_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_266_sumset__is__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_267_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_268_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_269_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_270_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_271_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_272_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_273_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_274_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_275_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_276_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_277_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_278_singletonD,axiom,
! [B4: nat,A: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_279_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_280_singleton__iff,axiom,
! [B4: nat,A: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_281_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_282_doubleton__eq__iff,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B4 @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_283_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_284_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_285_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_286_singleton__inject,axiom,
! [A: nat,B4: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B4 @ bot_bot_set_nat ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_287_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_288_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_289_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_290_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_291_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_292_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_293_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_294_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_295_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_296_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_297_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_298_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_299_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_300_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_301_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_302_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_303_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_304_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_305_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_306_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_307_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_308_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_309_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_310_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_311_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_312_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_313_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_314_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_315_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_316_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_317_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_318_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_319_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_320_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_321_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_322_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_323_finite__has__minimal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_324_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_325_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_326_finite__has__maximal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_327_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_328_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_329_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_330_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A6: set_nat,B3: nat] :
( ( A4
= ( insert_nat @ B3 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_331_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_332_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_333_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_334_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_335_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_336_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_337_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_338_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_339_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_340_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_341_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_342_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_343_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_344_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_345_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_346_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_347_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_348_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_349_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_350_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_351_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A6 )
=> ( member_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_352_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_353_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_354_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_355_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_356_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_357_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_358_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_359_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_360_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_361_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_362_insertE,axiom,
! [A: nat,B4: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_363_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_364_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_365_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_366_insertI2,axiom,
! [A: nat,B: set_nat,B4: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B4 @ B ) ) ) ).
% insertI2
thf(fact_367_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B7: set_a] :
( ( A2
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_368_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B7: set_nat] :
( ( A2
= ( insert_nat @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_369_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_370_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_371_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_372_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_373_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_374_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B4: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B4 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B4 @ C3 ) )
& ~ ( member_nat @ B4 @ C3 )
& ( B
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_375_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_376_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_377_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B7: set_nat] :
( ( A2
= ( insert_nat @ A @ B7 ) )
& ~ ( member_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_378_IntE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_379_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_380_IntD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_381_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_382_IntD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_383_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_384_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_385_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_386_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_387_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_388_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_389_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_390_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_391_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_392_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_393_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_394_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_395_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_396_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_397_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_398_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_399_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_400_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_401_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_402_card__1__singleton__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: a] :
( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_403_card__1__singleton__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_404_card__eq__SucD,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
=> ? [B2: a,B7: set_a] :
( ( A2
= ( insert_a @ B2 @ B7 ) )
& ~ ( member_a @ B2 @ B7 )
& ( ( finite_card_a @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_405_card__eq__SucD,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
=> ? [B2: nat,B7: set_nat] :
( ( A2
= ( insert_nat @ B2 @ B7 ) )
& ~ ( member_nat @ B2 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_406_card__Suc__eq,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A2
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_407_card__Suc__eq,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B3: nat,B6: set_nat] :
( ( A2
= ( insert_nat @ B3 @ B6 ) )
& ~ ( member_nat @ B3 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_408_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_409_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_410_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_411_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_412_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A4 )
| ( member_a @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_413_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A4 )
| ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_414_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_415_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_416_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_417_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A6 )
& ( member_a @ X2 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_418_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_419_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_420_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_421_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_422_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
| ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_423_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A6 )
| ( member_a @ X2 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_424_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_425_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_426_card__2__iff,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a,Y2: a] :
( ( S
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( X2 != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_427_card__2__iff,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat,Y2: nat] :
( ( S
= ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
& ( X2 != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_428_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_429_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_430_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_431_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_432_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_433_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_434_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_435_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_436_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_437_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_438_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_439_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_440_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_441_Int__insert__left,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) )
& ( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_442_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_443_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_444_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_445_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_446_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_447_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_448_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_449_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_450_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_451_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B9: set_a] :
( ( ord_less_eq_set_a @ B9 @ B )
=> ( C
!= ( sup_sup_set_a @ A8 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_452_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_453_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_454_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_455_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_456_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_457_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_458_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X2: nat] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_459_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X2: a] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_460_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_461_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_462_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MAT2968510229921005611_set_a @ bot_bot_set_nat @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_463_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_464_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MAT2968510229921005611_set_a @ bot_bot_set_nat @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_465_sumsetdiff__sing,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_466_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_467_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_468_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_469_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_470_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_471_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_472_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_473_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_474_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_475_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_476_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_477_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_478_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_479_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_480_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_481_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_482_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_483_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_484_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_485_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_486_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_487_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_488_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_489_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_490_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_491_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_492_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_493_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_494_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_495_le__inf__iff,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z4 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_496_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_497_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_498_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_499_le__sup__iff,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_nat @ X @ Z4 )
& ( ord_less_eq_nat @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_500_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_set_a @ X @ Z4 )
& ( ord_less_eq_set_a @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_501_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_502_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_503_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_504_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_505_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_506_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_507_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_508_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_509_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_510_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B4 ) )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_511_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_512_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_513_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_514_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_515_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_516_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_517_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_518_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_519_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_520_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_521_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_522_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_523_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_524_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_525_insert__Diff1,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_526_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_527_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_528_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_529_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_530_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_531_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_532_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_533_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_534_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_535_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_536_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_537_Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_538_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_539_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_540_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_541_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_542_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_543_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_544_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_545_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_546_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A6 )
& ~ ( member_a @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_547_diff__right__commute,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B4 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_548_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_549_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_550_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_551_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_552_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_553_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_554_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_555_Diff__mono,axiom,
! [A2: set_a,C: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_556_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_557_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_558_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_559_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_560_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_561_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_562_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_563_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_564_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_565_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_566_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_567_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_568_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_569_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_570_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_571_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_572_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_573_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_574_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_575_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_576_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_577_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_578_Diff__triv,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_579_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_580_Int__Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_581_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_582_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_583_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_584_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_585_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_586_Diff__Int,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_587_Diff__Un,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_588_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_589_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_590_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A7: set_nat] :
( ( X5 @ A7 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A7 )
& ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_591_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A7: set_a] :
( ( X5 @ A7 )
=> ? [X4: a] :
( ( member_a @ X4 @ A7 )
& ( ( X5 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_592_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_593_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_594_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_595_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_596_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_597_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_598_card__le__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_599_card__le__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_600_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_601_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_602_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_603_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_604_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z4 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) ) ) ).
% inf_assoc
thf(fact_605_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_606_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_607_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z4 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_608_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_609_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_610_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% sup_left_commute
thf(fact_611_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_612_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_613_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_614_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% sup_assoc
thf(fact_615_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_616_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_617_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% inf_sup_aci(6)
thf(fact_618_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% inf_sup_aci(7)
thf(fact_619_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_620_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_621_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_622_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_623_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_624_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_625_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_626_card_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ A2 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_627_card_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ A2 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_628_card_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_629_card_Oinsert__remove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_630_card__Suc__Diff1,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_631_card__Suc__Diff1,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_632_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_633_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_634_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_635_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_636_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_637_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_638_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_639_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_640_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_641_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_642_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_643_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_644_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_645_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_646_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_647_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_648_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_649_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_650_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_651_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_652_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_653_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_654_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_655_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_656_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_657_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_658_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_659_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_660_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_661_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_662_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_663_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_664_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_665_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_666_inf_OboundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_667_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_668_inf_OboundedI,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_669_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_670_inf__greatest,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_671_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_672_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_673_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_674_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_675_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_676_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_677_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_678_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_679_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_680_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_681_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_682_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_683_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_684_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_685_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_686_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_687_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_688_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_689_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_690_le__supE,axiom,
! [A: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_691_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_692_le__supI,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_693_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_694_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_695_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_696_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_697_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_698_le__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_699_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_700_le__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_701_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_702_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_703_sup_Omono,axiom,
! [C2: set_a,A: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_704_sup__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_705_sup__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_706_sup__least,axiom,
! [Y: nat,X: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_707_sup__least,axiom,
! [Y: set_a,X: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_708_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_709_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_710_sup_OorderE,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( A
= ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_711_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_712_sup_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( sup_sup_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% sup.orderI
thf(fact_713_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_714_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_715_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_716_sup_Oabsorb1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_717_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_718_sup_Oabsorb2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_719_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_720_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_721_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_722_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_723_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_724_sup_OboundedE,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B4 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_725_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_726_sup_OboundedI,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_727_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_728_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_729_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_730_sup_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_731_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_732_sup_Ocobounded2,axiom,
! [B4: nat,A: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_733_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_734_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_735_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_736_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_737_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_738_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_739_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_740_sup_OcoboundedI2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_741_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_742_sup__inf__distrib2,axiom,
! [Y: set_a,Z4: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z4 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z4 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_743_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% sup_inf_distrib1
thf(fact_744_inf__sup__distrib2,axiom,
! [Y: set_a,Z4: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z4 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_745_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z4 ) ) ) ).
% inf_sup_distrib1
thf(fact_746_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z4 ) ) ) ) ).
% distrib_imp2
thf(fact_747_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z4 ) ) ) ) ).
% distrib_imp1
thf(fact_748_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z4: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z4 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_749_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z4 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_750_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z4: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z4 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_751_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_752_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_753_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_754_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_755_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_756_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_757_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_758_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > num] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_759_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > num] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S2 )
=> ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_760_sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( collect_a
@ ^ [C4: a] :
? [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ g ) )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( inf_inf_set_a @ B @ g ) )
& ( C4
= ( addition @ X2 @ Y2 ) ) ) ) ) ) ).
% sumset_eq
thf(fact_761_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_762_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_763_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_764_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_765_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_766_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_767_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_768_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_769_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_770_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_771_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_772_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_773_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_774_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_775_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_776_bex__empty,axiom,
! [P: a > $o] :
~ ? [X4: a] :
( ( member_a @ X4 @ bot_bot_set_a )
& ( P @ X4 ) ) ).
% bex_empty
thf(fact_777_bex__empty,axiom,
! [P: nat > $o] :
~ ? [X4: nat] :
( ( member_nat @ X4 @ bot_bot_set_nat )
& ( P @ X4 ) ) ).
% bex_empty
thf(fact_778_finite__Collect__bex,axiom,
! [A2: set_a,Q: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
? [Y2: a] :
( ( member_a @ Y2 @ A2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y2: a] : ( Q @ Y2 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_779_finite__Collect__bex,axiom,
! [A2: set_a,Q: nat > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y2: a] :
( ( member_a @ Y2 @ A2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y2: nat] : ( Q @ Y2 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_780_finite__Collect__bex,axiom,
! [A2: set_nat,Q: a > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
? [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y2: a] : ( Q @ Y2 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_781_finite__Collect__bex,axiom,
! [A2: set_nat,Q: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
& ( Q @ X2 @ Y2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y2: nat] : ( Q @ Y2 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_782_Bex__def,axiom,
( bex_a
= ( ^ [A6: set_a,P2: a > $o] :
? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_783_Bex__def,axiom,
( bex_nat
= ( ^ [A6: set_nat,P2: nat > $o] :
? [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_784_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_785_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_786_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_787_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_788_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_789_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_790_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_791_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_792_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_793_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_794_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_795_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_796_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_797_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_798_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_799_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_800_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_801_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_802_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_803_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_804_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_805_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_806_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_807_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_808_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > num,C2: num] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_809_ord__le__eq__subst,axiom,
! [A: num,B4: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_810_ord__le__eq__subst,axiom,
! [A: num,B4: num,F: num > set_a,C2: set_a] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_811_ord__le__eq__subst,axiom,
! [A: num,B4: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_812_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_813_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_814_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_815_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_816_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_817_ord__eq__le__subst,axiom,
! [A: num,F: set_a > num,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_818_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B4: num,C2: num] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_819_ord__eq__le__subst,axiom,
! [A: set_a,F: num > set_a,B4: num,C2: num] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_820_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B4: num,C2: num] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_821_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_822_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_823_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_824_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_825_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_826_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_827_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_828_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_num @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_829_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_830_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_831_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > num,C2: num] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_num @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_832_order__subst2,axiom,
! [A: num,B4: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_833_order__subst2,axiom,
! [A: num,B4: num,F: num > set_a,C2: set_a] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_834_order__subst2,axiom,
! [A: num,B4: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ord_less_eq_num @ ( F @ B4 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_835_order__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_836_order__subst1,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_837_order__subst1,axiom,
! [A: nat,F: num > nat,B4: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_838_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_839_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_840_order__subst1,axiom,
! [A: set_a,F: num > set_a,B4: num,C2: num] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_841_order__subst1,axiom,
! [A: num,F: nat > num,B4: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_842_order__subst1,axiom,
! [A: num,F: set_a > num,B4: set_a,C2: set_a] :
( ( ord_less_eq_num @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_843_order__subst1,axiom,
! [A: num,F: num > num,B4: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_844_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_845_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_846_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_847_antisym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_848_antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_849_antisym,axiom,
! [A: num,B4: num] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ord_less_eq_num @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_850_dual__order_Otrans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_851_dual__order_Otrans,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_852_dual__order_Otrans,axiom,
! [B4: num,A: num,C2: num] :
( ( ord_less_eq_num @ B4 @ A )
=> ( ( ord_less_eq_num @ C2 @ B4 )
=> ( ord_less_eq_num @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_853_dual__order_Oantisym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_854_dual__order_Oantisym,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_855_dual__order_Oantisym,axiom,
! [B4: num,A: num] :
( ( ord_less_eq_num @ B4 @ A )
=> ( ( ord_less_eq_num @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_856_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_857_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_858_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_859_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_860_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B4: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_861_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_862_order__trans,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_863_order__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z4 )
=> ( ord_less_eq_num @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_864_order_Otrans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_865_order_Otrans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_866_order_Otrans,axiom,
! [A: num,B4: num,C2: num] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% order.trans
thf(fact_867_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_868_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_869_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_870_ord__le__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_871_ord__le__eq__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_872_ord__le__eq__trans,axiom,
! [A: num,B4: num,C2: num] :
( ( ord_less_eq_num @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_873_ord__eq__le__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_874_ord__eq__le__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( A = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_875_ord__eq__le__trans,axiom,
! [A: num,B4: num,C2: num] :
( ( A = B4 )
=> ( ( ord_less_eq_num @ B4 @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_876_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_877_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_878_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_879_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_880_le__cases3,axiom,
! [X: num,Y: num,Z4: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z4 ) )
=> ( ( ( ord_less_eq_num @ X @ Z4 )
=> ~ ( ord_less_eq_num @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z4 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z4 )
=> ~ ( ord_less_eq_num @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z4 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_881_nle__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_882_nle__le,axiom,
! [A: num,B4: num] :
( ( ~ ( ord_less_eq_num @ A @ B4 ) )
= ( ( ord_less_eq_num @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_883_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_884_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_885_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_886_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_887_card__Diff__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_888_card__Diff__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_889_diff__card__le__card__Diff,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_890_diff__card__le__card__Diff,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_891_card__Diff__subset__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_892_card__Diff__subset__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_893_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_894_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_895_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_896_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_897_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_898_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_899_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_900_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_901_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_902_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_903_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_904_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z4 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_905_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_906_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z4: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z4 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_907_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z4: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z4 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z4 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_908_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_909_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_910_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_911_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_912_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_913_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_914_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_915_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_916_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_917_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_918_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_919_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_920_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_921_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_922_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_923_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_924_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_925_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_926_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_927_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_928_verit__la__disequality,axiom,
! [A: nat,B4: nat] :
( ( A = B4 )
| ~ ( ord_less_eq_nat @ A @ B4 )
| ~ ( ord_less_eq_nat @ B4 @ A ) ) ).
% verit_la_disequality
thf(fact_929_verit__la__disequality,axiom,
! [A: num,B4: num] :
( ( A = B4 )
| ~ ( ord_less_eq_num @ A @ B4 )
| ~ ( ord_less_eq_num @ B4 @ A ) ) ).
% verit_la_disequality
thf(fact_930_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_931_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_932_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_933_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X2: a] :
( A6
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_934_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X2: nat] :
( A6
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_935_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_936_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_937_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_938_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ G ) ) ) )
& ( ~ ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_939_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) )
& ( ~ ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_940_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_941_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_942_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > num] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_num @ ( F @ ( lattic2897619205827179199_a_num @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_943_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > num] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_944_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X5: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_945_insert__subsetI,axiom,
! [X: a,A2: set_a,X5: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_946_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_947_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B4: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ A2 )
=> ( ( member_nat @ A @ G )
=> ( ( member_nat @ B4 @ B )
=> ( ( member_nat @ B4 @ G )
=> ( member_nat @ ( Addition @ A @ B4 ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_948_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_949_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_950_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
= ( ? [A4: nat,B3: nat] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_nat @ A4 @ A2 )
& ( member_nat @ A4 @ G )
& ( member_nat @ B3 @ B )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_951_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_952_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: nat,B2: nat] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ A3 @ G )
=> ( ( member_nat @ B2 @ B )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_953_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_954_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,B: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: nat,B2: nat] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_nat @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_955_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_956_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,B: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: nat,B3: nat] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_nat @ A4 @ G )
& ( B @ B3 )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_957_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_958_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,A: nat,B: nat > $o,B4: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_nat @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_nat @ B4 @ G )
=> ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_959_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_960_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_961_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_962_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_963_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_964_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_965_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_966_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_967_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_968_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_969_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_970_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_971_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_972_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_973_additive__abelian__group_Osumset__def,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition )
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( pluenn5670965976768739049tp_nat @ G @ Addition
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ) ).
% additive_abelian_group.sumset_def
thf(fact_974_additive__abelian__group_Osumset__def,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B6 ) ) ) ) ) ) ).
% additive_abelian_group.sumset_def
thf(fact_975_additive__abelian__group_Osumsetp__sumset__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumsetp_sumset_eq
thf(fact_976_additive__abelian__group_Osumsetp__sumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumsetp_sumset_eq
thf(fact_977_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_978_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_979_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_nat @ A2 @ G )
= bot_bot_set_nat )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_980_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_981_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_nat @ A2 @ G )
= bot_bot_set_nat )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_982_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_983_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B )
= bot_bot_set_nat )
= ( ( ( inf_inf_set_nat @ A2 @ G )
= bot_bot_set_nat )
| ( ( inf_inf_set_nat @ B @ G )
= bot_bot_set_nat ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_984_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_985_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_986_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_987_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_988_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_989_additive__abelian__group_Osumset__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B )
= ( collect_nat
@ ^ [C4: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ G ) )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( inf_inf_set_nat @ B @ G ) )
& ( C4
= ( Addition @ X2 @ Y2 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset_eq
thf(fact_990_additive__abelian__group_Osumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( collect_a
@ ^ [C4: a] :
? [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ G ) )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( inf_inf_set_a @ B @ G ) )
& ( C4
= ( Addition @ X2 @ Y2 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset_eq
thf(fact_991_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_992_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_993_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
& ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) )
| ( ( ( inf_inf_set_nat @ A2 @ G )
!= bot_bot_set_nat )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_994_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_995_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ Zero @ bot_bot_set_nat ) @ A2 )
= ( inf_inf_set_nat @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_996_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_997_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ Zero @ bot_bot_set_nat ) )
= ( inf_inf_set_nat @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_998_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_999_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_nat @ A2 @ G )
=> ( ( ord_less_eq_set_nat @ B @ G )
=> ( ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
| ( ( finite_card_nat @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_1000_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_1001_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( minus_minus_set_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ B @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_1002_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_1003_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ G ) )
= zero_zero_nat )
| ( ( finite_card_nat @ ( inf_inf_set_nat @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_1004_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_1005_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1006_prop__restrict,axiom,
! [X: a,Z5: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1007_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1008_Collect__restrict,axiom,
! [X5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1009_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( member_nat @ A @ G )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_1010_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_1011_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( nO_MAT4391312325019038905et_nat @ bot_bot_set_a @ A2 )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ X @ A2 ) @ B )
= ( sup_sup_set_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ X @ bot_bot_set_nat ) @ B ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_1012_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( nO_MAT2475032472373502585et_nat @ bot_bot_set_nat @ A2 )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ X @ A2 ) @ B )
= ( sup_sup_set_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ X @ bot_bot_set_nat ) @ B ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_1013_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_1014_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MAT2968510229921005611_set_a @ bot_bot_set_nat @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_1015_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,B: set_nat,A2: set_nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( nO_MAT4391312325019038905et_nat @ bot_bot_set_a @ B )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ X @ B ) )
= ( sup_sup_set_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_1016_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,B: set_nat,A2: set_nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( nO_MAT2475032472373502585et_nat @ bot_bot_set_nat @ B )
=> ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ X @ B ) )
= ( sup_sup_set_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_1017_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_1018_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MAT2968510229921005611_set_a @ bot_bot_set_nat @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_1019_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_1020_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_1021_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1022_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1023_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_1024_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1025_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_1026_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_1027_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_1028_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_1029_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_1030_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_1031_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_1032_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_1033_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_1034_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_1035_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_1036_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_1037_invertible__right__cancel,axiom,
! [X: a,Y: a,Z4: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z4 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z4 @ X ) )
= ( Y = Z4 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_1038_invertible__left__cancel,axiom,
! [X: a,Y: a,Z4: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z4 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z4 ) )
= ( Y = Z4 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_1039_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_1040_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_1041_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_1042_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_1043_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_1044_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_1045_Units__def,axiom,
( ( group_Units_a @ g @ addition @ zero )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ g )
& ( group_invertible_a @ g @ addition @ zero @ U2 ) ) ) ) ).
% Units_def
thf(fact_1046_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_1047_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_1048_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_1049_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_1050_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_1051_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_1052_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_1053_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_1054_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_1055_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_1056_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_1057_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_1058_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_1059_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_1060_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ X @ G )
=> ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_1061_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_1062_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_1063_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_1064_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_1065_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_1066_monoid_OsubgroupI,axiom,
! [M3: set_nat,Composition: nat > nat > nat,Unit: nat,G: set_nat] :
( ( group_monoid_nat @ M3 @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ G @ M3 )
=> ( ( member_nat @ Unit @ G )
=> ( ! [G3: nat,H: nat] :
( ( member_nat @ G3 @ G )
=> ( ( member_nat @ H @ G )
=> ( member_nat @ ( Composition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G )
=> ( group_invertible_nat @ M3 @ Composition @ Unit @ G3 ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G )
=> ( member_nat @ ( group_inverse_nat @ M3 @ Composition @ Unit @ G3 ) @ G ) )
=> ( group_subgroup_nat @ G @ M3 @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1067_monoid_OsubgroupI,axiom,
! [M3: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_monoid_a @ M3 @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G @ M3 )
=> ( ( member_a @ Unit @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ M3 @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ M3 @ Composition @ Unit @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ M3 @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1068_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_1069_image__eqI,axiom,
! [B4: set_a,F: a > set_a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1070_image__eqI,axiom,
! [B4: a,F: a > a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1071_image__eqI,axiom,
! [B4: nat,F: a > nat,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ B4 @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1072_image__eqI,axiom,
! [B4: a,F: nat > a,X: nat,A2: set_nat] :
( ( B4
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ B4 @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1073_image__eqI,axiom,
! [B4: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B4
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1074_image__ident,axiom,
! [Y6: set_a] :
( ( image_a_a
@ ^ [X2: a] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1075_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1076_image__is__empty,axiom,
! [F: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1077_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1078_image__is__empty,axiom,
! [F: nat > a,A2: set_nat] :
( ( ( image_nat_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1079_image__is__empty,axiom,
! [F: a > nat,A2: set_a] :
( ( ( image_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1080_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1081_empty__is__image,axiom,
! [F: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1082_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1083_empty__is__image,axiom,
! [F: nat > a,A2: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1084_empty__is__image,axiom,
! [F: a > nat,A2: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1085_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1086_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_1087_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1088_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1089_image__empty,axiom,
! [F: nat > a] :
( ( image_nat_a @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1090_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1091_finite__imageI,axiom,
! [F2: set_a,H3: a > set_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_set_a @ ( image_a_set_a @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1092_finite__imageI,axiom,
! [F2: set_a,H3: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1093_finite__imageI,axiom,
! [F2: set_a,H3: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1094_finite__imageI,axiom,
! [F2: set_nat,H3: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1095_finite__imageI,axiom,
! [F2: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1096_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1097_image__insert,axiom,
! [F: a > set_a,A: a,B: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A @ B ) )
= ( insert_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_insert
thf(fact_1098_image__insert,axiom,
! [F: a > a,A: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_1099_insert__image,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) )
= ( image_a_set_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1100_insert__image,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1101_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1102_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_nat_a @ F @ A2 ) )
= ( image_nat_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1103_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G4: nat > nat,S: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G4 @ X2 ) )
@ S )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_nat @ G4
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1104_if__image__distrib,axiom,
! [P: a > $o,F: a > set_a,G4: a > set_a,S: set_a] :
( ( image_a_set_a
@ ^ [X2: a] : ( if_set_a @ ( P @ X2 ) @ ( F @ X2 ) @ ( G4 @ X2 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_set_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1105_if__image__distrib,axiom,
! [P: nat > $o,F: nat > a,G4: nat > a,S: set_nat] :
( ( image_nat_a
@ ^ [X2: nat] : ( if_a @ ( P @ X2 ) @ ( F @ X2 ) @ ( G4 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_nat_a @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_a @ G4
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1106_if__image__distrib,axiom,
! [P: a > $o,F: a > a,G4: a > a,S: set_a] :
( ( image_a_a
@ ^ [X2: a] : ( if_a @ ( P @ X2 ) @ ( F @ X2 ) @ ( G4 @ X2 ) )
@ S )
= ( sup_sup_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1107_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1108_image__diff__subset,axiom,
! [F: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1109_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1110_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1111_all__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1112_all__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1113_all__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1114_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1115_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1116_ex__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1117_ex__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1118_ex__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1119_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1120_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1121_finite__subset__image,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_set_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1122_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A2: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1123_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A2: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1124_finite__subset__image,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1125_finite__surj,axiom,
! [A2: set_a,B: set_set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B ) ) ) ).
% finite_surj
thf(fact_1126_finite__surj,axiom,
! [A2: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1127_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1128_finite__surj,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1129_finite__surj,axiom,
! [A2: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1130_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_1131_image__Un,axiom,
! [F: a > set_a,A2: set_a,B: set_a] :
( ( image_a_set_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Un
thf(fact_1132_image__Un,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_1133_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_set_a @ ( image_a_set_a @ F @ A2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1134_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1135_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1136_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1137_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1138_imageE,axiom,
! [B4: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1139_imageE,axiom,
! [B4: a,F: a > a,A2: set_a] :
( ( member_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1140_imageE,axiom,
! [B4: a,F: nat > a,A2: set_nat] :
( ( member_a @ B4 @ ( image_nat_a @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1141_imageE,axiom,
! [B4: nat,F: a > nat,A2: set_a] :
( ( member_nat @ B4 @ ( image_a_nat @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1142_imageE,axiom,
! [B4: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1143_imageI,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_1144_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_1145_imageI,axiom,
! [X: a,A2: set_a,F: a > nat] :
( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1146_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_1147_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1148_image__iff,axiom,
! [Z4: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z4 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z4
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1149_image__iff,axiom,
! [Z4: a,F: a > a,A2: set_a] :
( ( member_a @ Z4 @ ( image_a_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z4
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1150_image__iff,axiom,
! [Z4: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z4
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1151_bex__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1152_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1153_bex__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1154_image__cong,axiom,
! [M3: set_a,N6: set_a,F: a > a,G4: a > a] :
( ( M3 = N6 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_a_a @ F @ M3 )
= ( image_a_a @ G4 @ N6 ) ) ) ) ).
% image_cong
thf(fact_1155_image__cong,axiom,
! [M3: set_a,N6: set_a,F: a > set_a,G4: a > set_a] :
( ( M3 = N6 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_a_set_a @ F @ M3 )
= ( image_a_set_a @ G4 @ N6 ) ) ) ) ).
% image_cong
thf(fact_1156_image__cong,axiom,
! [M3: set_nat,N6: set_nat,F: nat > nat,G4: nat > nat] :
( ( M3 = N6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G4 @ N6 ) ) ) ) ).
% image_cong
thf(fact_1157_ball__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1158_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1159_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1160_image__image,axiom,
! [F: set_a > a,G4: a > set_a,A2: set_a] :
( ( image_set_a_a @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G4 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1161_image__image,axiom,
! [F: set_a > set_a,G4: a > set_a,A2: set_a] :
( ( image_set_a_set_a @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G4 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1162_image__image,axiom,
! [F: a > a,G4: a > a,A2: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G4 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1163_image__image,axiom,
! [F: nat > nat,G4: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G4 @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G4 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1164_image__image,axiom,
! [F: a > set_a,G4: a > a,A2: set_a] :
( ( image_a_set_a @ F @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G4 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1165_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1166_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1167_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: nat,F: a > nat] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_nat @ B4 @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1168_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B4: a,F: nat > a] :
( ( member_nat @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1169_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B4: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1170_Compr__image__eq,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_set_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1171_Compr__image__eq,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1172_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1173_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1174_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1175_image__def,axiom,
( image_a_set_a
= ( ^ [F4: a > set_a,A6: set_a] :
( collect_set_a
@ ^ [Y2: set_a] :
? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( Y2
= ( F4 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1176_image__def,axiom,
( image_a_a
= ( ^ [F4: a > a,A6: set_a] :
( collect_a
@ ^ [Y2: a] :
? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( Y2
= ( F4 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1177_image__def,axiom,
( image_nat_nat
= ( ^ [F4: nat > nat,A6: set_nat] :
( collect_nat
@ ^ [Y2: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ( Y2
= ( F4 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1178_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1179_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1180_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1181_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1182_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1183_subset__image__iff,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1184_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1185_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1186_image__subset__iff,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1187_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1188_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B
!= ( image_a_set_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1189_subset__imageE,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1190_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1191_image__subsetI,axiom,
! [A2: set_a,F: a > nat,B: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1192_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1193_image__subsetI,axiom,
! [A2: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1194_image__subsetI,axiom,
! [A2: set_nat,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1195_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_1196_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1197_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1198_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > set_a,B: set_set_a] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1199_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > nat,B: set_nat] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1200_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1201_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B: set_a] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1202_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1203_image__Int__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1204_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1205_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1206_image__constant,axiom,
! [X: a,A2: set_a,C2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_set_a
@ ^ [X2: a] : C2
@ A2 )
= ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ).
% image_constant
thf(fact_1207_image__constant,axiom,
! [X: a,A2: set_a,C2: a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_a
@ ^ [X2: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1208_image__constant,axiom,
! [X: nat,A2: set_nat,C2: a] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1209_image__constant,axiom,
! [X: a,A2: set_a,C2: nat] :
( ( member_a @ X @ A2 )
=> ( ( image_a_nat
@ ^ [X2: a] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1210_image__constant,axiom,
! [X: nat,A2: set_nat,C2: nat] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1211_image__constant__conv,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X2: a] : C2
@ A2 )
= ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1212_image__constant__conv,axiom,
! [A2: set_a,C2: a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1213_image__constant__conv,axiom,
! [A2: set_a,C2: nat] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1214_image__constant__conv,axiom,
! [A2: set_nat,C2: a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1215_image__constant__conv,axiom,
! [A2: set_nat,C2: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1216_the__elem__image__unique,axiom,
! [A2: set_a,F: a > a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1217_the__elem__image__unique,axiom,
! [A2: set_a,F: a > set_a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1218_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > nat,X: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1219_card__image__le,axiom,
! [A2: set_nat,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1220_card__image__le,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1221_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1222_sumset,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( insert_a @ ( addition @ A4 @ B3 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ g ) ) )
@ ( inf_inf_set_a @ A2 @ g ) ) ) ) ).
% sumset
thf(fact_1223_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1224_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_1225_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1226_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_1227_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_1228_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_1229_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_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__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_1232_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_1233_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_1234_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1235_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1236_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_1237_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1238_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1239_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_1240_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1241_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1242_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1243_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_1244_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1245_binomial__mono,axiom,
! [K: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K2 ) ) ) ) ).
% binomial_mono
thf(fact_1246_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_1247_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1248_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1249_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1250_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1251_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1252_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_1253_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_1254_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1255_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1256_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1257_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_1258_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_1259_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1260_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1261_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1262_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1263_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1264_bezout1__nat,axiom,
! [A: nat,B4: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B4 )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B4 @ Y3 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B4 @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_1265_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1266_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1267_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1268_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1269_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__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 ) ).
thf(help_If_3_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ aa @ aa ) ) @ ( binomial @ ( suc @ ( finite_card_a @ aa ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
%------------------------------------------------------------------------------