TPTP Problem File: SLH0225^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0014_Set_Multiset_Extras/prob_00067_002281__27880354_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1354 ( 585 unt; 87 typ; 0 def)
% Number of atoms : 3283 (1110 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9829 ( 385 ~; 56 |; 183 &;7787 @)
% ( 0 <=>;1418 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 258 ( 258 >; 0 *; 0 +; 0 <<)
% Number of symbols : 79 ( 78 usr; 13 con; 0-3 aty)
% Number of variables : 3292 ( 243 ^;2960 !; 89 ?;3292 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:42:23.386
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (78)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > 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__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_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
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_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
sup_sup_multiset_nat: multiset_nat > multiset_nat > multiset_nat ).
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_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Multiset_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: multiset_nat > $o ).
thf(sy_c_Multiset_Ois__empty_001tf__a,type,
is_empty_a: multiset_a > $o ).
thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
mset_set_nat: set_nat > multiset_nat ).
thf(sy_c_Multiset_Omset__set_001tf__a,type,
mset_set_a: set_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Oreplicate__mset_001tf__a,type,
replicate_mset_a: nat > a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Multiset__Order_Omultp_092_060_094sub_062H_092_060_094sub_062O_001tf__a,type,
multiset_multp_H_O_a: ( a > a > $o ) > multiset_a > multiset_a > $o ).
thf(sy_c_Multiset__Order_Opreorder__class_Oless__multiset_092_060_094sub_062H_092_060_094sub_062O_001t__Nat__Onat,type,
multis7733847888720353750_O_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
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__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > 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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: multiset_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_C____,type,
c: set_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1265)
thf(fact_0_count__inject,axiom,
! [X: multiset_nat,Y: multiset_nat] :
( ( ( count_nat @ X )
= ( count_nat @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_1_count__inject,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( count_set_a @ X )
= ( count_set_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_2_count__inject,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( count_a @ X )
= ( count_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_3_multiset__eqI,axiom,
! [A: multiset_nat,B: multiset_nat] :
( ! [X2: nat] :
( ( count_nat @ A @ X2 )
= ( count_nat @ B @ X2 ) )
=> ( A = B ) ) ).
% multiset_eqI
thf(fact_4_multiset__eqI,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ! [X2: set_a] :
( ( count_set_a @ A @ X2 )
= ( count_set_a @ B @ X2 ) )
=> ( A = B ) ) ).
% multiset_eqI
thf(fact_5_multiset__eqI,axiom,
! [A: multiset_a,B: multiset_a] :
( ! [X2: a] :
( ( count_a @ A @ X2 )
= ( count_a @ B @ X2 ) )
=> ( A = B ) ) ).
% multiset_eqI
thf(fact_6_multiset__eq__iff,axiom,
( ( ^ [Y2: multiset_nat,Z: multiset_nat] : ( Y2 = Z ) )
= ( ^ [M: multiset_nat,N: multiset_nat] :
! [A2: nat] :
( ( count_nat @ M @ A2 )
= ( count_nat @ N @ A2 ) ) ) ) ).
% multiset_eq_iff
thf(fact_7_multiset__eq__iff,axiom,
( ( ^ [Y2: multiset_set_a,Z: multiset_set_a] : ( Y2 = Z ) )
= ( ^ [M: multiset_set_a,N: multiset_set_a] :
! [A2: set_a] :
( ( count_set_a @ M @ A2 )
= ( count_set_a @ N @ A2 ) ) ) ) ).
% multiset_eq_iff
thf(fact_8_multiset__eq__iff,axiom,
( ( ^ [Y2: multiset_a,Z: multiset_a] : ( Y2 = Z ) )
= ( ^ [M: multiset_a,N: multiset_a] :
! [A2: a] :
( ( count_a @ M @ A2 )
= ( count_a @ N @ A2 ) ) ) ) ).
% multiset_eq_iff
thf(fact_9_cdef,axiom,
( ( minus_minus_set_a @ b @ ( set_mset_a @ a2 ) )
= c ) ).
% cdef
thf(fact_10_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_11_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_12_zero__reorient,axiom,
! [X: multiset_nat] :
( ( zero_z7348594199698428585et_nat = X )
= ( X = zero_z7348594199698428585et_nat ) ) ).
% zero_reorient
thf(fact_13_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_14_un,axiom,
( ( sup_sup_set_a @ c @ ( set_mset_a @ a2 ) )
= b ) ).
% un
thf(fact_15_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_16_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ A3 @ A3 )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_17_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A3 @ A3 )
= zero_z7348594199698428585et_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_18_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_19_diff__zero,axiom,
! [A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ A3 @ zero_zero_multiset_a )
= A3 ) ).
% diff_zero
thf(fact_20_diff__zero,axiom,
! [A3: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A3 @ zero_z7348594199698428585et_nat )
= A3 ) ).
% diff_zero
thf(fact_21_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_22_zero__diff,axiom,
! [A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A3 )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_23_zero__diff,axiom,
! [A3: multiset_nat] :
( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ A3 )
= zero_z7348594199698428585et_nat ) ).
% zero_diff
thf(fact_24_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_25_count__empty,axiom,
! [A3: nat] :
( ( count_nat @ zero_z7348594199698428585et_nat @ A3 )
= zero_zero_nat ) ).
% count_empty
thf(fact_26_count__empty,axiom,
! [A3: set_a] :
( ( count_set_a @ zero_z5079479921072680283_set_a @ A3 )
= zero_zero_nat ) ).
% count_empty
thf(fact_27_count__empty,axiom,
! [A3: a] :
( ( count_a @ zero_zero_multiset_a @ A3 )
= zero_zero_nat ) ).
% count_empty
thf(fact_28_assms_I1_J,axiom,
finite_finite_a @ b ).
% assms(1)
thf(fact_29_assms_I2_J,axiom,
ord_less_eq_set_a @ ( set_mset_a @ a2 ) @ b ).
% assms(2)
thf(fact_30_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: multiset_a,C2: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A3 @ C2 ) @ B2 )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A3 @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_31_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: multiset_nat,C2: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A3 @ C2 ) @ B2 )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A3 @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_32_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C2 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_33_multiset__nonemptyE,axiom,
! [A: multiset_set_a] :
( ( A != zero_z5079479921072680283_set_a )
=> ~ ! [X2: set_a] :
~ ( member_set_a @ X2 @ ( set_mset_set_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_34_multiset__nonemptyE,axiom,
! [A: multiset_nat] :
( ( A != zero_z7348594199698428585et_nat )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ ( set_mset_nat @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_35_multiset__nonemptyE,axiom,
! [A: multiset_a] :
( ( A != zero_zero_multiset_a )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ ( set_mset_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_36_count__eq__zero__iff,axiom,
! [M2: multiset_nat,X: nat] :
( ( ( count_nat @ M2 @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_mset_nat @ M2 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_37_count__eq__zero__iff,axiom,
! [M2: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M2 @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M2 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_38_count__eq__zero__iff,axiom,
! [M2: multiset_a,X: a] :
( ( ( count_a @ M2 @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M2 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_39_count__inI,axiom,
! [M2: multiset_nat,X: nat] :
( ( ( count_nat @ M2 @ X )
!= zero_zero_nat )
=> ( member_nat @ X @ ( set_mset_nat @ M2 ) ) ) ).
% count_inI
thf(fact_40_count__inI,axiom,
! [M2: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M2 @ X )
!= zero_zero_nat )
=> ( member_set_a @ X @ ( set_mset_set_a @ M2 ) ) ) ).
% count_inI
thf(fact_41_count__inI,axiom,
! [M2: multiset_a,X: a] :
( ( ( count_a @ M2 @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M2 ) ) ) ).
% count_inI
thf(fact_42_zero__multiset_Orep__eq,axiom,
( ( count_nat @ zero_z7348594199698428585et_nat )
= ( ^ [A2: nat] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_43_zero__multiset_Orep__eq,axiom,
( ( count_set_a @ zero_z5079479921072680283_set_a )
= ( ^ [A2: set_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_44_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A2: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_45_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_46_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_47_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_48_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_49_fin,axiom,
finite_finite_a @ ( set_mset_a @ a2 ) ).
% fin
thf(fact_50_UnCI,axiom,
! [C2: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_51_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_52_UnCI,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_53_Un__iff,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C2 @ A )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_54_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_55_Un__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_56_DiffI,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_57_DiffI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_58_DiffI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_59_Diff__iff,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C2 @ A )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_60_Diff__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_61_Diff__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_62_Diff__idemp,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_63_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_64_sup_Oidem,axiom,
! [A3: nat] :
( ( sup_sup_nat @ A3 @ A3 )
= A3 ) ).
% sup.idem
thf(fact_65_sup_Oidem,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ A3 )
= A3 ) ).
% sup.idem
thf(fact_66_sup_Oidem,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% sup.idem
thf(fact_67_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_68_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_69_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_70_sup_Oleft__idem,axiom,
! [A3: nat,B2: nat] :
( ( sup_sup_nat @ A3 @ ( sup_sup_nat @ A3 @ B2 ) )
= ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.left_idem
thf(fact_71_sup_Oleft__idem,axiom,
! [A3: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) )
= ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.left_idem
thf(fact_72_sup_Oleft__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B2 ) )
= ( sup_sup_set_a @ A3 @ B2 ) ) ).
% sup.left_idem
thf(fact_73_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_74_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_75_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_76_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_77_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_78_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_79_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_80_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_81_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_82_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_83_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_90_sup_Oright__idem,axiom,
! [A3: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A3 @ B2 ) @ B2 )
= ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.right_idem
thf(fact_91_sup_Oright__idem,axiom,
! [A3: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.right_idem
thf(fact_92_sup_Oright__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A3 @ B2 ) ) ).
% sup.right_idem
thf(fact_93_count__diff,axiom,
! [M2: multiset_set_a,N3: multiset_set_a,A3: set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ M2 @ N3 ) @ A3 )
= ( minus_minus_nat @ ( count_set_a @ M2 @ A3 ) @ ( count_set_a @ N3 @ A3 ) ) ) ).
% count_diff
thf(fact_94_count__diff,axiom,
! [M2: multiset_nat,N3: multiset_nat,A3: nat] :
( ( count_nat @ ( minus_8522176038001411705et_nat @ M2 @ N3 ) @ A3 )
= ( minus_minus_nat @ ( count_nat @ M2 @ A3 ) @ ( count_nat @ N3 @ A3 ) ) ) ).
% count_diff
thf(fact_95_count__diff,axiom,
! [M2: multiset_a,N3: multiset_a,A3: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M2 @ N3 ) @ A3 )
= ( minus_minus_nat @ ( count_a @ M2 @ A3 ) @ ( count_a @ N3 @ A3 ) ) ) ).
% count_diff
thf(fact_96_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_97_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_98_sup_Obounded__iff,axiom,
! [B2: multiset_nat,C2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ B2 @ C2 ) @ A3 )
= ( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
& ( ord_le6602235886369790592et_nat @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_99_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq_set_a @ B2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_100_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_101_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_102_le__sup__iff,axiom,
! [X: multiset_nat,Y: multiset_nat,Z2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ X @ Y ) @ Z2 )
= ( ( ord_le6602235886369790592et_nat @ X @ Z2 )
& ( ord_le6602235886369790592et_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_103_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_104_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_105_finite__set__mset,axiom,
! [M2: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M2 ) ) ).
% finite_set_mset
thf(fact_106_finite__set__mset,axiom,
! [M2: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M2 ) ) ).
% finite_set_mset
thf(fact_107_finite__set__mset,axiom,
! [M2: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M2 ) ) ).
% finite_set_mset
thf(fact_108_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C3 )
= ( ( ord_less_eq_set_nat @ A @ C3 )
& ( ord_less_eq_set_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_109_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 )
= ( ( ord_less_eq_set_a @ A @ C3 )
& ( ord_less_eq_set_a @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_110_diff__empty,axiom,
! [M2: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M2 @ zero_zero_multiset_a )
= M2 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M2 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_111_diff__empty,axiom,
! [M2: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ M2 @ zero_z7348594199698428585et_nat )
= M2 )
& ( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ M2 )
= zero_z7348594199698428585et_nat ) ) ).
% diff_empty
thf(fact_112_Multiset_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_113_Multiset_Odiff__cancel,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ A )
= zero_z7348594199698428585et_nat ) ).
% Multiset.diff_cancel
thf(fact_114_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_115_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_116_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_117_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_118_subset__trans,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_119_subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_120_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_121_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_122_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_123_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_124_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_125_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_126_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_127_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_128_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_129_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_130_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_131_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_132_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_133_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_134_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_135_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_136_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_137_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_138_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_139_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_140_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_141_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_142_minus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa: multiset_set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ X @ Xa ) )
= ( ^ [A2: set_a] : ( minus_minus_nat @ ( count_set_a @ X @ A2 ) @ ( count_set_a @ Xa @ A2 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_143_minus__multiset_Orep__eq,axiom,
! [X: multiset_nat,Xa: multiset_nat] :
( ( count_nat @ ( minus_8522176038001411705et_nat @ X @ Xa ) )
= ( ^ [A2: nat] : ( minus_minus_nat @ ( count_nat @ X @ A2 ) @ ( count_nat @ Xa @ A2 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_144_minus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X @ Xa ) )
= ( ^ [A2: a] : ( minus_minus_nat @ ( count_a @ X @ A2 ) @ ( count_a @ Xa @ A2 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_145_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_146_sup_OcoboundedI2,axiom,
! [C2: multiset_nat,B2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ C2 @ B2 )
=> ( ord_le6602235886369790592et_nat @ C2 @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_147_sup_OcoboundedI2,axiom,
! [C2: set_a,B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_148_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_149_sup_OcoboundedI1,axiom,
! [C2: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A3 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_150_sup_OcoboundedI1,axiom,
! [C2: multiset_nat,A3: multiset_nat,B2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ C2 @ A3 )
=> ( ord_le6602235886369790592et_nat @ C2 @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_151_sup_OcoboundedI1,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_152_sup_OcoboundedI1,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_153_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A2 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_154_sup_Oabsorb__iff2,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [A2: multiset_nat,B4: multiset_nat] :
( ( sup_sup_multiset_nat @ A2 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_155_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( sup_sup_set_a @ A2 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_156_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( sup_sup_nat @ A2 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_157_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ B4 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_158_sup_Oabsorb__iff1,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [B4: multiset_nat,A2: multiset_nat] :
( ( sup_sup_multiset_nat @ A2 @ B4 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_159_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( sup_sup_set_a @ A2 @ B4 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_160_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( sup_sup_nat @ A2 @ B4 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_161_sup_Ocobounded2,axiom,
! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_162_sup_Ocobounded2,axiom,
! [B2: multiset_nat,A3: multiset_nat] : ( ord_le6602235886369790592et_nat @ B2 @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_163_sup_Ocobounded2,axiom,
! [B2: set_a,A3: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_164_sup_Ocobounded2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_165_sup_Ocobounded1,axiom,
! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_166_sup_Ocobounded1,axiom,
! [A3: multiset_nat,B2: multiset_nat] : ( ord_le6602235886369790592et_nat @ A3 @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_167_sup_Ocobounded1,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_168_sup_Ocobounded1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_169_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( A2
= ( sup_sup_set_nat @ A2 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_170_sup_Oorder__iff,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [B4: multiset_nat,A2: multiset_nat] :
( A2
= ( sup_sup_multiset_nat @ A2 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_171_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A2: set_a] :
( A2
= ( sup_sup_set_a @ A2 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_172_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( A2
= ( sup_sup_nat @ A2 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_173_sup_OboundedI,axiom,
! [B2: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ A3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_174_sup_OboundedI,axiom,
! [B2: multiset_nat,A3: multiset_nat,C2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
=> ( ( ord_le6602235886369790592et_nat @ C2 @ A3 )
=> ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ B2 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_175_sup_OboundedI,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_176_sup_OboundedI,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_177_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ~ ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_178_sup_OboundedE,axiom,
! [B2: multiset_nat,C2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
=> ~ ( ord_le6602235886369790592et_nat @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_179_sup_OboundedE,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ~ ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_180_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A3 )
=> ~ ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_181_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_182_sup__absorb2,axiom,
! [X: multiset_nat,Y: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ X @ Y )
=> ( ( sup_sup_multiset_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_183_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_184_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_185_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_186_sup__absorb1,axiom,
! [Y: multiset_nat,X: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ Y @ X )
=> ( ( sup_sup_multiset_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_187_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_188_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_189_sup_Oabsorb2,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_190_sup_Oabsorb2,axiom,
! [A3: multiset_nat,B2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A3 @ B2 )
=> ( ( sup_sup_multiset_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_191_sup_Oabsorb2,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_192_sup_Oabsorb2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_193_sup_Oabsorb1,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_194_sup_Oabsorb1,axiom,
! [B2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
=> ( ( sup_sup_multiset_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_195_sup_Oabsorb1,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_196_sup_Oabsorb1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_197_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_198_sup__unique,axiom,
! [F: multiset_nat > multiset_nat > multiset_nat,X: multiset_nat,Y: multiset_nat] :
( ! [X2: multiset_nat,Y3: multiset_nat] : ( ord_le6602235886369790592et_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: multiset_nat,Y3: multiset_nat] : ( ord_le6602235886369790592et_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: multiset_nat,Y3: multiset_nat,Z3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ Y3 @ X2 )
=> ( ( ord_le6602235886369790592et_nat @ Z3 @ X2 )
=> ( ord_le6602235886369790592et_nat @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_multiset_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_199_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_200_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_201_sup_OorderI,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_202_sup_OorderI,axiom,
! [A3: multiset_nat,B2: multiset_nat] :
( ( A3
= ( sup_sup_multiset_nat @ A3 @ B2 ) )
=> ( ord_le6602235886369790592et_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_203_sup_OorderI,axiom,
! [A3: set_a,B2: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_204_sup_OorderI,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_205_sup_OorderE,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( A3
= ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_206_sup_OorderE,axiom,
! [B2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
=> ( A3
= ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_207_sup_OorderE,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( A3
= ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_208_sup_OorderE,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( A3
= ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_209_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_210_le__iff__sup,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [X3: multiset_nat,Y4: multiset_nat] :
( ( sup_sup_multiset_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_211_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_212_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_213_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_214_sup__least,axiom,
! [Y: multiset_nat,X: multiset_nat,Z2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ Y @ X )
=> ( ( ord_le6602235886369790592et_nat @ Z2 @ X )
=> ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_215_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_216_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_217_sup__mono,axiom,
! [A3: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_218_sup__mono,axiom,
! [A3: multiset_nat,C2: multiset_nat,B2: multiset_nat,D: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A3 @ C2 )
=> ( ( ord_le6602235886369790592et_nat @ B2 @ D )
=> ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ A3 @ B2 ) @ ( sup_sup_multiset_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_219_sup__mono,axiom,
! [A3: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_220_sup__mono,axiom,
! [A3: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_221_sup_Omono,axiom,
! [C2: set_nat,A3: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A3 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D ) @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_222_sup_Omono,axiom,
! [C2: multiset_nat,A3: multiset_nat,D: multiset_nat,B2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ C2 @ A3 )
=> ( ( ord_le6602235886369790592et_nat @ D @ B2 )
=> ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ C2 @ D ) @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_223_sup_Omono,axiom,
! [C2: set_a,A3: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_224_sup_Omono,axiom,
! [C2: nat,A3: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_225_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_226_le__supI2,axiom,
! [X: multiset_nat,B2: multiset_nat,A3: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ X @ B2 )
=> ( ord_le6602235886369790592et_nat @ X @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_227_le__supI2,axiom,
! [X: set_a,B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_228_le__supI2,axiom,
! [X: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_229_le__supI1,axiom,
! [X: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_230_le__supI1,axiom,
! [X: multiset_nat,A3: multiset_nat,B2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ X @ A3 )
=> ( ord_le6602235886369790592et_nat @ X @ ( sup_sup_multiset_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_231_le__supI1,axiom,
! [X: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_232_le__supI1,axiom,
! [X: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A3 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_233_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_234_sup__ge2,axiom,
! [Y: multiset_nat,X: multiset_nat] : ( ord_le6602235886369790592et_nat @ Y @ ( sup_sup_multiset_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_235_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_236_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_237_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_238_sup__ge1,axiom,
! [X: multiset_nat,Y: multiset_nat] : ( ord_le6602235886369790592et_nat @ X @ ( sup_sup_multiset_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_239_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_240_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_241_le__supI,axiom,
! [A3: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_242_le__supI,axiom,
! [A3: multiset_nat,X: multiset_nat,B2: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ A3 @ X )
=> ( ( ord_le6602235886369790592et_nat @ B2 @ X )
=> ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ A3 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_243_le__supI,axiom,
! [A3: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_244_le__supI,axiom,
! [A3: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_245_le__supE,axiom,
! [A3: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_246_le__supE,axiom,
! [A3: multiset_nat,B2: multiset_nat,X: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ ( sup_sup_multiset_nat @ A3 @ B2 ) @ X )
=> ~ ( ( ord_le6602235886369790592et_nat @ A3 @ X )
=> ~ ( ord_le6602235886369790592et_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_247_le__supE,axiom,
! [A3: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A3 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_248_le__supE,axiom,
! [A3: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A3 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_249_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_250_inf__sup__ord_I3_J,axiom,
! [X: multiset_nat,Y: multiset_nat] : ( ord_le6602235886369790592et_nat @ X @ ( sup_sup_multiset_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_251_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_252_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_253_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_254_inf__sup__ord_I4_J,axiom,
! [Y: multiset_nat,X: multiset_nat] : ( ord_le6602235886369790592et_nat @ Y @ ( sup_sup_multiset_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_255_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_256_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_257_double__diff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_258_double__diff,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_259_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_260_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_261_Diff__mono,axiom,
! [A: set_nat,C3: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_262_Diff__mono,axiom,
! [A: set_a,C3: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_263_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_264_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_265_subset__UnE,axiom,
! [C3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C3
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_266_subset__UnE,axiom,
! [C3: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C3
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_267_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_268_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_269_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_270_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_271_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_272_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_273_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_274_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_275_Un__least,axiom,
! [A: set_nat,C3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_276_Un__least,axiom,
! [A: set_a,C3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_277_Un__mono,axiom,
! [A: set_nat,C3: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_278_Un__mono,axiom,
! [A: set_a,C3: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_279_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_280_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_281_diffs0__imp__equal,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M3 )
= zero_zero_nat )
=> ( M3 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_282_in__diffD,axiom,
! [A3: set_a,M2: multiset_set_a,N3: multiset_set_a] :
( ( member_set_a @ A3 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M2 @ N3 ) ) )
=> ( member_set_a @ A3 @ ( set_mset_set_a @ M2 ) ) ) ).
% in_diffD
thf(fact_283_in__diffD,axiom,
! [A3: nat,M2: multiset_nat,N3: multiset_nat] :
( ( member_nat @ A3 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M2 @ N3 ) ) )
=> ( member_nat @ A3 @ ( set_mset_nat @ M2 ) ) ) ).
% in_diffD
thf(fact_284_in__diffD,axiom,
! [A3: a,M2: multiset_a,N3: multiset_a] :
( ( member_a @ A3 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N3 ) ) )
=> ( member_a @ A3 @ ( set_mset_a @ M2 ) ) ) ).
% in_diffD
thf(fact_285_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C3 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_286_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C3 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_287_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_288_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_289_sup__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_290_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_291_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_292_sup_Oleft__commute,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A3 @ C2 ) )
= ( sup_sup_nat @ A3 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_293_sup_Oleft__commute,axiom,
! [B2: set_nat,A3: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ C2 ) )
= ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_294_sup_Oleft__commute,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A3 @ C2 ) )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_295_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_296_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_297_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_298_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A2: nat,B4: nat] : ( sup_sup_nat @ B4 @ A2 ) ) ) ).
% sup.commute
thf(fact_299_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A2 ) ) ) ).
% sup.commute
thf(fact_300_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A2 ) ) ) ).
% sup.commute
thf(fact_301_sup__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_302_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_303_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_304_sup_Oassoc,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A3 @ B2 ) @ C2 )
= ( sup_sup_nat @ A3 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_305_sup_Oassoc,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_306_sup_Oassoc,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_307_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_308_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_309_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_310_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_311_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_312_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_313_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_314_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_315_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_316_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_317_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_318_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_319_DiffD2,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_320_DiffD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_321_DiffD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_322_DiffD1,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_323_DiffD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_324_DiffD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_325_DiffE,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_326_DiffE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_327_DiffE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_328_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C3 ) ) ) ).
% Un_left_commute
thf(fact_329_Un__left__commute,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C3 ) ) ) ).
% Un_left_commute
thf(fact_330_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_331_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_332_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_333_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_334_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_335_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_336_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C3 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_337_Un__assoc,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_338_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_339_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_340_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_341_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_342_UnI2,axiom,
! [C2: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_343_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_344_UnI2,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_345_UnI1,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_346_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_347_UnI1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_348_UnE,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_349_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_350_UnE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_351_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C3 ) @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_352_Un__Diff,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C3 ) @ ( minus_minus_set_a @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_353_finite__Un,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_354_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_355_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_356_finite__Diff2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( finite_finite_set_a @ A ) ) ) ).
% finite_Diff2
thf(fact_357_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_358_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_359_finite__Diff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_360_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_361_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_362_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_363_dual__order_Orefl,axiom,
! [A3: multiset_nat] : ( ord_le6602235886369790592et_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_364_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_365_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_366_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_367_order__refl,axiom,
! [X: multiset_nat] : ( ord_le6602235886369790592et_nat @ X @ X ) ).
% order_refl
thf(fact_368_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_369_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_370_Multiset_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: multiset_a] : ( A4 = zero_zero_multiset_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_371_Multiset_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: multiset_nat] : ( A4 = zero_z7348594199698428585et_nat ) ) ) ).
% Multiset.is_empty_def
thf(fact_372_infinite__Un,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_373_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_374_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_375_Un__infinite,axiom,
! [S: set_set_a,T2: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_376_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_377_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_378_finite__UnI,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_379_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_380_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_381_Diff__infinite__finite,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_382_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_383_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_384_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_385_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_386_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_387_diff__is__0__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_388_diff__is__0__eq_H,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_389_eq__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_390_le__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_391_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_392_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_393_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_394_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_395_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_396_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_397_Multiset_Odiff__right__commute,axiom,
! [M2: multiset_a,N3: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ N3 ) @ Q )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ Q ) @ N3 ) ) ).
% Multiset.diff_right_commute
thf(fact_398_Multiset_Odiff__right__commute,axiom,
! [M2: multiset_nat,N3: multiset_nat,Q: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ N3 ) @ Q )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ Q ) @ N3 ) ) ).
% Multiset.diff_right_commute
thf(fact_399_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_400_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_401_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_402_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_403_nle__le,axiom,
! [A3: multiset_nat,B2: multiset_nat] :
( ( ~ ( ord_le6602235886369790592et_nat @ A3 @ B2 ) )
= ( ( ord_le6602235886369790592et_nat @ B2 @ A3 )
& ( B2 != A3 ) ) ) ).
% nle_le
thf(fact_404_nle__le,axiom,
! [A3: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A3 )
& ( B2 != A3 ) ) ) ).
% nle_le
thf(fact_405_le__cases3,axiom,
! [X: multiset_nat,Y: multiset_nat,Z2: multiset_nat] :
( ( ( ord_le6602235886369790592et_nat @ X @ Y )
=> ~ ( ord_le6602235886369790592et_nat @ Y @ Z2 ) )
=> ( ( ( ord_le6602235886369790592et_nat @ Y @ X )
=> ~ ( ord_le6602235886369790592et_nat @ X @ Z2 ) )
=> ( ( ( ord_le6602235886369790592et_nat @ X @ Z2 )
=> ~ ( ord_le6602235886369790592et_nat @ Z2 @ Y ) )
=> ( ( ( ord_le6602235886369790592et_nat @ Z2 @ Y )
=> ~ ( ord_le6602235886369790592et_nat @ Y @ X ) )
=> ( ( ( ord_le6602235886369790592et_nat @ Y @ Z2 )
=> ~ ( ord_le6602235886369790592et_nat @ Z2 @ X ) )
=> ~ ( ( ord_le6602235886369790592et_nat @ Z2 @ X )
=> ~ ( ord_le6602235886369790592et_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_406_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_407_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_408_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: multiset_nat,Z: multiset_nat] : ( Y2 = Z ) )
= ( ^ [X3: multiset_nat,Y4: multiset_nat] :
( ( ord_le6602235886369790592et_nat @ X3 @ Y4 )
& ( ord_le6602235886369790592et_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_409_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_410_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_411_ord__eq__le__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( A3 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_412_ord__eq__le__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( A3 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_413_ord__le__eq__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_414_ord__le__eq__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_415_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_416_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_417_order_Otrans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_418_order_Otrans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_419_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_420_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_421_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_422_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_423_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_424_dual__order_Oantisym,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_425_dual__order_Oantisym,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_426_dual__order_Otrans,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_427_dual__order_Otrans,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_428_antisym,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_429_antisym,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_430_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_431_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_432_order__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_433_order__subst1,axiom,
! [A3: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_434_order__subst1,axiom,
! [A3: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_435_order__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_436_order__subst2,axiom,
! [A3: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_437_order__subst2,axiom,
! [A3: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_438_order__subst2,axiom,
! [A3: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_439_order__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_440_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_441_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_442_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_443_ord__eq__le__subst,axiom,
! [A3: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_444_ord__eq__le__subst,axiom,
! [A3: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_445_ord__eq__le__subst,axiom,
! [A3: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_446_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_447_ord__le__eq__subst,axiom,
! [A3: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
! [A3: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_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_452_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_453_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_454_finite__has__maximal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A3 @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_455_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_456_finite__has__minimal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A3 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_457_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_458_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_459_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_460_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_461_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_462_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_463_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_464_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_465_disj,axiom,
( ( inf_inf_set_a @ c @ ( set_mset_a @ a2 ) )
= bot_bot_set_a ) ).
% disj
thf(fact_466_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_467_mset__set__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( mset_set_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_8522176038001411705et_nat @ ( mset_set_nat @ A ) @ ( mset_set_nat @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_468_mset__set__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( mset_set_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_3765977307040488491iset_a @ ( mset_set_a @ A ) @ ( mset_set_a @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_469_count__greater__zero__iff,axiom,
! [M2: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M2 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M2 ) ) ) ).
% count_greater_zero_iff
thf(fact_470_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_471_card__Diff__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_472_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_473_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_474_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_475_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_476_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_477_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_478_inf_Oleft__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% inf.left_idem
thf(fact_479_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_480_inf_Oright__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% inf.right_idem
thf(fact_481_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_482_IntI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_483_Int__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_484_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_485_inf_Obounded__iff,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_486_inf_Obounded__iff,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_487_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_488_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_489_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_490_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_491_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_492_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_493_sup__bot_Oright__neutral,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_494_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A3 @ B2 ) )
= ( ( A3 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_495_sup__bot_Oleft__neutral,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_496_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A3 @ B2 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_497_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_498_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_499_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_500_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_501_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_502_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_503_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_504_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_505_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_506_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_507_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_508_Int__subset__iff,axiom,
! [C3: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C3 @ A )
& ( ord_less_eq_set_a @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_509_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_510_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_511_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_512_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_513_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_514_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_515_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_516_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_517_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_518_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_519_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_520_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_521_mset__set__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ( mset_set_a @ A )
= ( mset_set_a @ B ) )
= ( A = B ) ) ) ) ).
% mset_set_eq_iff
thf(fact_522_mset__set__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ( mset_set_nat @ A )
= ( mset_set_nat @ B ) )
= ( A = B ) ) ) ) ).
% mset_set_eq_iff
thf(fact_523_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_524_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_525_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_526_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_527_zero__less__diff,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M3 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% zero_less_diff
thf(fact_528_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_529_set__mset__eq__empty__iff,axiom,
! [M2: multiset_a] :
( ( ( set_mset_a @ M2 )
= bot_bot_set_a )
= ( M2 = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_530_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_531_mset__set_Oempty,axiom,
( ( mset_set_a @ bot_bot_set_a )
= zero_zero_multiset_a ) ).
% mset_set.empty
thf(fact_532_mset__set_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( mset_set_a @ A )
= zero_zero_multiset_a ) ) ).
% mset_set.infinite
thf(fact_533_mset__set_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( mset_set_nat @ A )
= zero_z7348594199698428585et_nat ) ) ).
% mset_set.infinite
thf(fact_534_finite__set__mset__mset__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_mset_nat @ ( mset_set_nat @ A ) )
= A ) ) ).
% finite_set_mset_mset_set
thf(fact_535_finite__set__mset__mset__set,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( set_mset_a @ ( mset_set_a @ A ) )
= A ) ) ).
% finite_set_mset_mset_set
thf(fact_536_elem__mset__set,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ ( set_mset_nat @ ( mset_set_nat @ A ) ) )
= ( member_nat @ X @ A ) ) ) ).
% elem_mset_set
thf(fact_537_elem__mset__set,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ ( set_mset_a @ ( mset_set_a @ A ) ) )
= ( member_a @ X @ A ) ) ) ).
% elem_mset_set
thf(fact_538_count__mset__set_I3_J,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( count_a @ ( mset_set_a @ A ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_539_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_540_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_541_count__mset__set_I2_J,axiom,
! [A: set_nat,X: nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( count_nat @ ( mset_set_nat @ A ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(2)
thf(fact_542_count__mset__set_I2_J,axiom,
! [A: set_a,X: a] :
( ~ ( finite_finite_a @ A )
=> ( ( count_a @ ( mset_set_a @ A ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(2)
thf(fact_543_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_544_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_545_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_546_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_547_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_548_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_549_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_550_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_551_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_552_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_553_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_554_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_555_card__gt__0__iff,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
= ( ( A != bot_bot_set_a )
& ( finite_finite_a @ A ) ) ) ).
% card_gt_0_iff
thf(fact_556_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_557_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_558_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_559_order_Oasym,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A3 ) ) ).
% order.asym
thf(fact_560_ord__eq__less__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( A3 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_561_ord__less__eq__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_562_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_563_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_564_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_565_dual__order_Oasym,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B2 ) ) ).
% dual_order.asym
thf(fact_566_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_567_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_568_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_569_order_Ostrict__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_570_bot_Oextremum__strict,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_571_bot_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_572_bot_Onot__eq__extremum,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_573_bot_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_574_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_575_dual__order_Ostrict__trans,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_576_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( A3 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_577_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( A3 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_578_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_579_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_580_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_581_order__less__asym_H,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A3 ) ) ).
% order_less_asym'
thf(fact_582_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_583_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_584_ord__less__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_585_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_586_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_587_order__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_588_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_589_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_590_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_591_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_592_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_593_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_594_equal__card__inter__fin__eq__sets,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( ( ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) )
= ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ) ).
% equal_card_inter_fin_eq_sets
thf(fact_595_equal__card__inter__fin__eq__sets,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ B ) )
= ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ) ).
% equal_card_inter_fin_eq_sets
thf(fact_596_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_597_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_598_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_599_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_600_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_601_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_602_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_603_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_604_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( P @ M5 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_605_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_606_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_607_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_608_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_609_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_610_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_611_inf_Oassoc,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_612_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_613_less__infI1,axiom,
! [A3: set_a,X: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_614_less__infI1,axiom,
! [A3: nat,X: nat,B2: nat] :
( ( ord_less_nat @ A3 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_615_less__infI2,axiom,
! [B2: set_a,X: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_616_less__infI2,axiom,
! [B2: nat,X: nat,A3: nat] :
( ( ord_less_nat @ B2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_617_inf_Oabsorb3,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb3
thf(fact_618_inf_Oabsorb3,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb3
thf(fact_619_inf_Oabsorb4,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_620_inf_Oabsorb4,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_621_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_622_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_623_inf_Oleft__commute,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A3 @ C2 ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_624_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_625_inf_Ostrict__boundedE,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A3 @ B2 )
=> ~ ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_626_inf_Ostrict__boundedE,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_627_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B4 ) )
& ( A2 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_628_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B4 ) )
& ( A2 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_629_inf_Ostrict__coboundedI1,axiom,
! [A3: set_a,C2: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_630_inf_Ostrict__coboundedI1,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A3 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_631_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_632_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_633_IntE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_634_IntD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_635_IntD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_636_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_637_equals0D,axiom,
! [A: set_a,A3: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A ) ) ).
% equals0D
thf(fact_638_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_639_Int__assoc,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C3 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_640_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_641_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_642_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_643_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_644_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_645_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_646_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_647_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_648_Int__left__commute,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C3 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C3 ) ) ) ).
% Int_left_commute
thf(fact_649_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_650_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_651_card__ge__0__finite,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
=> ( finite_finite_a @ A ) ) ).
% card_ge_0_finite
thf(fact_652_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_653_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_654_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_655_card__less__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_656_card__less__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_657_mset__set__empty__iff,axiom,
! [A: set_nat] :
( ( ( mset_set_nat @ A )
= zero_z7348594199698428585et_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% mset_set_empty_iff
thf(fact_658_mset__set__empty__iff,axiom,
! [A: set_a] :
( ( ( mset_set_a @ A )
= zero_zero_multiset_a )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% mset_set_empty_iff
thf(fact_659_infinite__set__mset__mset__set,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( set_mset_nat @ ( mset_set_nat @ A ) )
= bot_bot_set_nat ) ) ).
% infinite_set_mset_mset_set
thf(fact_660_infinite__set__mset__mset__set,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( set_mset_a @ ( mset_set_a @ A ) )
= bot_bot_set_a ) ) ).
% infinite_set_mset_mset_set
thf(fact_661_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_662_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_663_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_664_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_665_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_666_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_667_inf_OcoboundedI2,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_668_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_669_inf_OcoboundedI1,axiom,
! [A3: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_670_inf_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_671_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( inf_inf_set_a @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_672_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_673_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_674_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( inf_inf_nat @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_675_inf_Ocobounded2,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_676_inf_Ocobounded2,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_677_inf_Ocobounded1,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_678_inf_Ocobounded1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_679_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_680_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( A2
= ( inf_inf_nat @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_681_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_682_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_683_inf_OboundedI,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_684_inf_OboundedI,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_685_inf_OboundedE,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ~ ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_686_inf_OboundedE,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_687_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_688_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_689_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_690_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_691_inf_Oabsorb2,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_692_inf_Oabsorb2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_693_inf_Oabsorb1,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_694_inf_Oabsorb1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_695_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_696_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_697_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_698_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_699_inf_OorderI,axiom,
! [A3: set_a,B2: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B2 ) )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_700_inf_OorderI,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_701_inf_OorderE,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_702_inf_OorderE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( A3
= ( inf_inf_nat @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_703_le__infI2,axiom,
! [B2: set_a,X: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_704_le__infI2,axiom,
! [B2: nat,X: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_705_le__infI1,axiom,
! [A3: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_706_le__infI1,axiom,
! [A3: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_707_inf__mono,axiom,
! [A3: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_708_inf__mono,axiom,
! [A3: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_709_le__infI,axiom,
! [X: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_710_le__infI,axiom,
! [X: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A3 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_711_le__infE,axiom,
! [X: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A3 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_712_le__infE,axiom,
! [X: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A3 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_713_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_714_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_715_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_716_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_717_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_718_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_719_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_720_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_721_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_722_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_723_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_724_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_725_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_726_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_727_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_728_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_729_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_730_order__less__le__subst2,axiom,
! [A3: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_731_order__less__le__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_732_order__less__le__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_733_order__less__le__subst1,axiom,
! [A3: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_734_order__less__le__subst1,axiom,
! [A3: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_set_a @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_735_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_736_order__le__less__subst2,axiom,
! [A3: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_737_order__le__less__subst2,axiom,
! [A3: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_738_order__le__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_739_order__le__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_740_order__le__less__subst1,axiom,
! [A3: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_741_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_742_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_743_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_744_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_745_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_746_order__neq__le__trans,axiom,
! [A3: set_a,B2: set_a] :
( ( A3 != B2 )
=> ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ord_less_set_a @ A3 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_747_order__neq__le__trans,axiom,
! [A3: nat,B2: nat] :
( ( A3 != B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_748_order__le__neq__trans,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_a @ A3 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_749_order__le__neq__trans,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_750_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_751_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_752_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_753_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_754_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_755_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_756_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_757_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_758_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ A3 )
=> ( ord_less_eq_set_a @ B2 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_759_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_760_order_Ostrict__implies__order,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_761_order_Ostrict__implies__order,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_762_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_763_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_764_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( ord_less_set_a @ B2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_765_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_766_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( ord_less_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_767_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_768_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_769_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_770_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( ord_less_set_a @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_771_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_nat @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_772_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_773_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_774_order_Ostrict__trans2,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_775_order_Ostrict__trans2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_776_order_Ostrict__trans1,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_777_order_Ostrict__trans1,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_778_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_779_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_780_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_set_a @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_781_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_782_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_783_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_784_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_785_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_786_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_787_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_788_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_789_nless__le,axiom,
! [A3: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A3 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ).
% nless_le
thf(fact_790_nless__le,axiom,
! [A3: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A3 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ).
% nless_le
thf(fact_791_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_792_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_793_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_794_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_795_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_796_gr__implies__not__zero,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_797_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_798_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_799_Int__greatest,axiom,
! [C3: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( ( ord_less_eq_set_a @ C3 @ B )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_800_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_801_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_802_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_803_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_804_Int__mono,axiom,
! [A: set_a,C3: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_805_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_806_Diff__Int__distrib,axiom,
! [C3: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A ) @ ( inf_inf_set_a @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_807_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_808_Diff__Int2,axiom,
! [A: set_a,C3: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ ( inf_inf_set_a @ B @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_809_Int__Diff,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C3 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_810_Un__Int__distrib2,axiom,
! [B: set_a,C3: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C3 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C3 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_811_Int__Un__distrib2,axiom,
! [B: set_a,C3: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C3 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C3 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_812_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_813_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_814_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C3 ) ) @ ( inf_inf_set_a @ C3 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C3 ) ) @ ( sup_sup_set_a @ C3 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_815_sup_Ostrict__coboundedI2,axiom,
! [C2: set_a,B2: set_a,A3: set_a] :
( ( ord_less_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_816_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A3: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_817_sup_Ostrict__coboundedI1,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( ord_less_set_a @ C2 @ A3 )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_818_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A3 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_819_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B4 ) )
& ( A2 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_820_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B4 ) )
& ( A2 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_821_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_set_a @ B2 @ A3 )
=> ~ ( ord_less_set_a @ C2 @ A3 ) ) ) ).
% sup.strict_boundedE
thf(fact_822_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_nat @ B2 @ A3 )
=> ~ ( ord_less_nat @ C2 @ A3 ) ) ) ).
% sup.strict_boundedE
thf(fact_823_sup_Oabsorb4,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_824_sup_Oabsorb4,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_825_sup_Oabsorb3,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_set_a @ B2 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb3
thf(fact_826_sup_Oabsorb3,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb3
thf(fact_827_less__supI2,axiom,
! [X: set_a,B2: set_a,A3: set_a] :
( ( ord_less_set_a @ X @ B2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% less_supI2
thf(fact_828_less__supI2,axiom,
! [X: nat,B2: nat,A3: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% less_supI2
thf(fact_829_less__supI1,axiom,
! [X: set_a,A3: set_a,B2: set_a] :
( ( ord_less_set_a @ X @ A3 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% less_supI1
thf(fact_830_less__supI1,axiom,
! [X: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ X @ A3 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% less_supI1
thf(fact_831_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_832_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_833_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_834_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_835_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_836_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_837_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_838_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_839_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_840_diff__less__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_841_diff__less__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_842_less__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_843_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_844_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_845_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_846_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_847_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_848_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_849_card__Diff__subset__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_850_card__Diff__subset__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_851_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_852_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_853_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_854_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_855_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_856_infinite__arbitrarily__large,axiom,
! [A: set_a,N2: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N2 )
& ( ord_less_eq_set_a @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_857_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_858_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_859_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_860_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C3: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_861_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_862_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S )
=> ( ( ( finite_card_a @ T4 )
= N2 )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_863_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C3 )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_864_exists__subset__between,axiom,
! [A: set_a,N2: nat,C3: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C3 ) )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( finite_finite_a @ C3 )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A @ B7 )
& ( ord_less_eq_set_a @ B7 @ C3 )
& ( ( finite_card_a @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_865_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_866_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_867_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_868_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_869_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_870_card__le__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_871_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C3 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_872_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_873_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_874_Diff__Int,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C3 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C3 ) ) ) ).
% Diff_Int
thf(fact_875_Diff__Un,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C3 ) ) ) ).
% Diff_Un
thf(fact_876_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_877_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_878_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_879_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_880_diff__less,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ) ) ).
% diff_less
thf(fact_881_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_882_diff__card__le__card__Diff,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_883_in__diff__count,axiom,
! [A3: a,M2: multiset_a,N3: multiset_a] :
( ( member_a @ A3 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N3 ) ) )
= ( ord_less_nat @ ( count_a @ N3 @ A3 ) @ ( count_a @ M2 @ A3 ) ) ) ).
% in_diff_count
thf(fact_884_mset__set__set__mset__empty__mempty,axiom,
! [D2: multiset_a] :
( ( ( mset_set_a @ ( set_mset_a @ D2 ) )
= zero_zero_multiset_a )
= ( D2 = zero_zero_multiset_a ) ) ).
% mset_set_set_mset_empty_mempty
thf(fact_885_set__card__diff__ge__zero,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( A != B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_886_set__card__diff__ge__zero,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( A != B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_887_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_888_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_889_card__inter__lt__single,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_inter_lt_single
thf(fact_890_card__inter__lt__single,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) @ ( finite_card_a @ A ) ) ) ) ).
% card_inter_lt_single
thf(fact_891_card__subset__not__gt__card,axiom,
! [A: set_nat,Ps: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ Ps ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ A ) ) ) ).
% card_subset_not_gt_card
thf(fact_892_card__subset__not__gt__card,axiom,
! [A: set_a,Ps: set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ Ps ) )
=> ~ ( ord_less_eq_set_a @ Ps @ A ) ) ) ).
% card_subset_not_gt_card
thf(fact_893_mset__nempty__set__nempty,axiom,
! [A: multiset_a] :
( ( A != zero_zero_multiset_a )
= ( ( set_mset_a @ A )
!= bot_bot_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_894_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_895_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_896_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_897_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_898_finite__psubset__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ! [B8: set_a] :
( ( ord_less_set_a @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_899_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_900_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_901_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_902_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_903_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_set_a @ A @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_904_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_905_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C3 )
=> ( ord_less_set_a @ A @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_906_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_907_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_908_set__count__size__min,axiom,
! [N2: nat,A: multiset_a,A3: a] :
( ( ord_less_eq_nat @ N2 @ ( count_a @ A @ A3 ) )
=> ( ord_less_eq_nat @ N2 @ ( size_size_multiset_a @ A ) ) ) ).
% set_count_size_min
thf(fact_909_mset__size__ne0__set__card,axiom,
! [A: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ A ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_mset_a @ A ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_910_psubset__card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A @ B )
=> ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_911_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_912_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A3: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A3 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_913_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A3: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_914_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A3: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A3 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_915_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A3: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A3 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_916_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_917_card__psubset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_918_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_919_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_920_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_921_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_922_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_923_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_924_set__diff__non__empty__not__subset,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C3 ) )
=> ( ( C3 != bot_bot_set_a )
=> ( ( A != bot_bot_set_a )
=> ( ( B != bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ A @ C3 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_925_elem__exists__non__empty__set,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ A ) ) ).
% elem_exists_non_empty_set
thf(fact_926_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_927_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_928_less__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M: multiset_nat,N: multiset_nat] :
( ( M != N )
& ! [Y4: nat] :
( ( ord_less_nat @ ( count_nat @ N @ Y4 ) @ ( count_nat @ M @ Y4 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y4 @ X3 )
& ( ord_less_nat @ ( count_nat @ M @ X3 ) @ ( count_nat @ N @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O
thf(fact_929_less__eq__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [M: multiset_nat,N: multiset_nat] :
! [Y4: nat] :
( ( ord_less_nat @ ( count_nat @ N @ Y4 ) @ ( count_nat @ M @ Y4 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y4 @ X3 )
& ( ord_less_nat @ ( count_nat @ M @ X3 ) @ ( count_nat @ N @ X3 ) ) ) ) ) ) ).
% less_eq_multiset\<^sub>H\<^sub>O
thf(fact_930_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X5 )
& ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_931_psubsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_932_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_933_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_934_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_935_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_936_ex__gt__imp__less__multiset,axiom,
! [N3: multiset_nat,M2: multiset_nat] :
( ? [Y5: nat] :
( ( member_nat @ Y5 @ ( set_mset_nat @ N3 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ( ord_less_nat @ X2 @ Y5 ) ) )
=> ( ord_le5777773500796000884et_nat @ M2 @ N3 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_937_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_938_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_939_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_940_less__multiset_092_060_094sub_062H_092_060_094sub_062O__def,axiom,
( multis7733847888720353750_O_nat
= ( ^ [M: multiset_nat,N: multiset_nat] :
( ( M != N )
& ! [Y4: nat] :
( ( ord_less_nat @ ( count_nat @ N @ Y4 ) @ ( count_nat @ M @ Y4 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y4 @ X3 )
& ( ord_less_nat @ ( count_nat @ M @ X3 ) @ ( count_nat @ N @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O_def
thf(fact_941_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_942_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X6: a] :
( ( member_a @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_943_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_944_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_945_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_946_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_947_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% minf(8)
thf(fact_948_minf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T3 @ X6 ) ) ).
% minf(7)
thf(fact_949_minf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T3 ) ) ).
% minf(5)
thf(fact_950_minf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T3 ) ) ).
% minf(4)
thf(fact_951_minf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T3 ) ) ).
% minf(3)
thf(fact_952_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_953_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_954_pinf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T3 @ X6 ) ) ).
% pinf(7)
thf(fact_955_pinf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T3 ) ) ).
% pinf(5)
thf(fact_956_pinf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(4)
thf(fact_957_pinf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(3)
thf(fact_958_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_959_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_960_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% pinf(6)
thf(fact_961_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% pinf(8)
thf(fact_962_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% minf(6)
thf(fact_963_complete__interval,axiom,
! [A3: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A3 @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A3 @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A3 @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_964_verit__comp__simplify1_I3_J,axiom,
! [B10: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B10 @ A8 ) )
= ( ord_less_nat @ A8 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_965_verit__la__disequality,axiom,
! [A3: nat,B2: nat] :
( ( A3 = B2 )
| ~ ( ord_less_eq_nat @ A3 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ).
% verit_la_disequality
thf(fact_966_verit__comp__simplify1_I2_J,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_967_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_968_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_969_multp_092_060_094sub_062H_092_060_094sub_062O__def,axiom,
( multiset_multp_H_O_a
= ( ^ [R2: a > a > $o,M: multiset_a,N: multiset_a] :
( ( M != N )
& ! [Y4: a] :
( ( ord_less_nat @ ( count_a @ N @ Y4 ) @ ( count_a @ M @ Y4 ) )
=> ? [X3: a] :
( ( R2 @ Y4 @ X3 )
& ( ord_less_nat @ ( count_a @ M @ X3 ) @ ( count_a @ N @ X3 ) ) ) ) ) ) ) ).
% multp\<^sub>H\<^sub>O_def
thf(fact_970_size__mset__removeAll__mset__le__iff,axiom,
! [M2: multiset_a,X: a] :
( ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( replicate_mset_a @ ( count_a @ M2 @ X ) @ X ) ) ) @ ( size_size_multiset_a @ M2 ) )
= ( member_a @ X @ ( set_mset_a @ M2 ) ) ) ).
% size_mset_removeAll_mset_le_iff
thf(fact_971_ex__gt__count__imp__le__multiset,axiom,
! [M2: multiset_set_a,N3: multiset_set_a,X: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ M2 @ N3 ) ) )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ( ord_less_nat @ ( count_set_a @ M2 @ X ) @ ( count_set_a @ N3 @ X ) )
=> ( ord_le5765082015083327056_set_a @ M2 @ N3 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_972_ex__gt__count__imp__le__multiset,axiom,
! [M2: multiset_nat,N3: multiset_nat,X: nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M2 @ N3 ) ) )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ord_less_nat @ ( count_nat @ M2 @ X ) @ ( count_nat @ N3 @ X ) )
=> ( ord_le5777773500796000884et_nat @ M2 @ N3 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_973_set__mset__minus__replicate__mset_I2_J,axiom,
! [N2: nat,A: multiset_a,A3: a] :
( ( ord_less_nat @ N2 @ ( count_a @ A @ A3 ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ A @ ( replicate_mset_a @ N2 @ A3 ) ) )
= ( set_mset_a @ A ) ) ) ).
% set_mset_minus_replicate_mset(2)
thf(fact_974_add__left__cancel,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_975_add__right__cancel,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_976_add__le__cancel__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_977_add__le__cancel__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_978_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_979_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_980_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_981_add__cancel__right__right,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_982_add__cancel__right__left,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ A3 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_983_add__cancel__left__right,axiom,
! [A3: nat,B2: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_984_add__cancel__left__left,axiom,
! [B2: nat,A3: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_985_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_986_add__less__cancel__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_987_add__less__cancel__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_988_add__diff__cancel__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_989_add__diff__cancel__left_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_990_add__diff__cancel__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_991_add__diff__cancel__right_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_992_le__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_993_le__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_994_add__le__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_995_add__le__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_996_add__less__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_997_add__less__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_998_less__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_999_less__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1000_diff__add__zero,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1001_set__mset__union,axiom,
! [M2: multiset_a,N3: multiset_a] :
( ( set_mset_a @ ( plus_plus_multiset_a @ M2 @ N3 ) )
= ( sup_sup_set_a @ ( set_mset_a @ M2 ) @ ( set_mset_a @ N3 ) ) ) ).
% set_mset_union
thf(fact_1002_count__replicate__mset,axiom,
! [Y: a,X: a,N2: nat] :
( ( ( Y = X )
=> ( ( count_a @ ( replicate_mset_a @ N2 @ X ) @ Y )
= N2 ) )
& ( ( Y != X )
=> ( ( count_a @ ( replicate_mset_a @ N2 @ X ) @ Y )
= zero_zero_nat ) ) ) ).
% count_replicate_mset
thf(fact_1003_in__replicate__mset,axiom,
! [X: a,N2: nat,Y: a] :
( ( member_a @ X @ ( set_mset_a @ ( replicate_mset_a @ N2 @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( X = Y ) ) ) ).
% in_replicate_mset
thf(fact_1004_ex__replicate__mset__if__all__elems__eq,axiom,
! [M2: multiset_a,Y: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( X2 = Y ) )
=> ? [N5: nat] :
( M2
= ( replicate_mset_a @ N5 @ Y ) ) ) ).
% ex_replicate_mset_if_all_elems_eq
thf(fact_1005_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_1006_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1007_add__implies__diff,axiom,
! [C2: nat,B2: nat,A3: nat] :
( ( ( plus_plus_nat @ C2 @ B2 )
= A3 )
=> ( C2
= ( minus_minus_nat @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1008_diff__diff__eq,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C2 )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1009_union__iff,axiom,
! [A3: a,A: multiset_a,B: multiset_a] :
( ( member_a @ A3 @ ( set_mset_a @ ( plus_plus_multiset_a @ A @ B ) ) )
= ( ( member_a @ A3 @ ( set_mset_a @ A ) )
| ( member_a @ A3 @ ( set_mset_a @ B ) ) ) ) ).
% union_iff
thf(fact_1010_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1011_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1012_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1013_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1014_add_Oassoc,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_1015_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B4: nat] : ( plus_plus_nat @ B4 @ A2 ) ) ) ).
% add.commute
thf(fact_1016_add_Oleft__commute,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_1017_add__left__imp__eq,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_1018_add__right__imp__eq,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_1019_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1020_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1021_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1022_add__strict__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1023_add__strict__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1024_add__strict__right__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1025_add__less__imp__less__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1026_add__less__imp__less__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1027_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1028_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1029_add__mono__thms__linordered__semiring_I1_J,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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1030_add__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1031_add__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1032_less__eqE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A3 @ C4 ) ) ) ).
% less_eqE
thf(fact_1033_add__right__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_1034_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% le_iff_add
thf(fact_1035_add__le__imp__le__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1036_add__le__imp__le__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1037_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_1038_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1039_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1040_add__nonpos__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1041_add__nonneg__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1042_add__increasing2,axiom,
! [C2: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1043_add__decreasing2,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1044_add__increasing,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1045_add__decreasing,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1046_add__less__le__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1047_add__le__less__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1048_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1049_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1050_add__neg__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1051_add__pos__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1052_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ! [C4: nat] :
( ( B2
= ( plus_plus_nat @ A3 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1053_pos__add__strict,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1054_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus_nat @ C2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1055_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B2 @ A3 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1056_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1057_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1058_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1059_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A3 )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1060_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1061_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1062_le__add__diff,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_1063_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1064_add__neg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1065_add__nonneg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1066_add__nonpos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1067_add__pos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1068_add__strict__increasing,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1069_add__strict__increasing2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1070_removeAll__notin,axiom,
! [A3: a,A: multiset_a] :
( ~ ( member_a @ A3 @ ( set_mset_a @ A ) )
=> ( ( minus_3765977307040488491iset_a @ A @ ( replicate_mset_a @ ( count_a @ A @ A3 ) @ A3 ) )
= A ) ) ).
% removeAll_notin
thf(fact_1071_mset__set__Union,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( mset_set_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( plus_p6334493942879108393et_nat @ ( mset_set_nat @ A ) @ ( mset_set_nat @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_1072_mset__set__Union,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( mset_set_a @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_multiset_a @ ( mset_set_a @ A ) @ ( mset_set_a @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_1073_le__add__diff__inverse,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1074_le__add__diff__inverse2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1075_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1076_add__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1077_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1078_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1079_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1080_add__gr__0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1081_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1082_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1083_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1084_count__union,axiom,
! [M2: multiset_a,N3: multiset_a,A3: a] :
( ( count_a @ ( plus_plus_multiset_a @ M2 @ N3 ) @ A3 )
= ( plus_plus_nat @ ( count_a @ M2 @ A3 ) @ ( count_a @ N3 @ A3 ) ) ) ).
% count_union
thf(fact_1085_add__eq__self__zero,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= M3 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1086_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1087_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1088_diff__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_cancel2
thf(fact_1089_diff__add__inverse,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M3 ) @ N2 )
= M3 ) ).
% diff_add_inverse
thf(fact_1090_diff__add__inverse2,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N2 ) @ N2 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1091_add__leE,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1092_le__add1,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) ) ).
% le_add1
thf(fact_1093_le__add2,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M3 @ N2 ) ) ).
% le_add2
thf(fact_1094_add__leD1,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% add_leD1
thf(fact_1095_add__leD2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1096_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1097_add__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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1098_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1099_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1100_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1101_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1102_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1103_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1104_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1105_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1106_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1107_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1108_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1109_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1110_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1111_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M6: nat,N5: nat] :
( ( ord_less_nat @ M6 @ N5 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1112_diff__add__0,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1113_add__diff__inverse__nat,axiom,
! [M3: nat,N2: nat] :
( ~ ( ord_less_nat @ M3 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M3 @ N2 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1114_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1115_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1116_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1117_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1118_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1119_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1120_plus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa: multiset_a] :
( ( count_a @ ( plus_plus_multiset_a @ X @ Xa ) )
= ( ^ [A2: a] : ( plus_plus_nat @ ( count_a @ X @ A2 ) @ ( count_a @ Xa @ A2 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_1121_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1122_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ( ( ord_less_nat @ A3 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1123_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1124_card__Un__le,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ).
% card_Un_le
thf(fact_1125_count__in__diffI,axiom,
! [N3: multiset_a,X: a,M2: multiset_a] :
( ! [N5: nat] :
( ( count_a @ N3 @ X )
!= ( plus_plus_nat @ N5 @ ( count_a @ M2 @ X ) ) )
=> ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N3 ) ) ) ) ).
% count_in_diffI
thf(fact_1126_card__Un__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1127_card__Un__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A @ B ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1128_card__Un__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1129_card__Un__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( finite_card_a @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1130_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1131_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1132_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ~ ( ord_less_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1133_set__mset__minus__replicate__mset_I1_J,axiom,
! [A: multiset_a,A3: a,N2: nat] :
( ( ord_less_eq_nat @ ( count_a @ A @ A3 ) @ N2 )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ A @ ( replicate_mset_a @ N2 @ A3 ) ) )
= ( minus_minus_set_a @ ( set_mset_a @ A ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% set_mset_minus_replicate_mset(1)
thf(fact_1134_insertCI,axiom,
! [A3: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A3 @ B )
=> ( A3 = B2 ) )
=> ( member_a @ A3 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_1135_insert__iff,axiom,
! [A3: a,B2: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B2 @ A ) )
= ( ( A3 = B2 )
| ( member_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_1136_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1137_finite__insert,axiom,
! [A3: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A3 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_1138_finite__insert,axiom,
! [A3: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1139_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_1140_Int__insert__left__if0,axiom,
! [A3: a,C3: set_a,B: set_a] :
( ~ ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C3 )
= ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1141_Int__insert__left__if1,axiom,
! [A3: a,C3: set_a,B: set_a] :
( ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C3 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1142_insert__inter__insert,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1143_Int__insert__right__if0,axiom,
! [A3: a,A: set_a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1144_Int__insert__right__if1,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1145_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1146_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1147_Un__insert__right,axiom,
! [A: set_a,A3: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1148_Un__insert__left,axiom,
! [A3: a,B: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A3 @ B ) @ C3 )
= ( insert_a @ A3 @ ( sup_sup_set_a @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1149_singleton__insert__inj__eq,axiom,
! [B2: a,A3: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A3 @ A ) )
= ( ( A3 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1150_singleton__insert__inj__eq_H,axiom,
! [A3: a,A: set_a,B2: a] :
( ( ( insert_a @ A3 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A3 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1151_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1152_disjoint__insert_I1_J,axiom,
! [B: set_a,A3: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A3 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1153_insert__disjoint_I2_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B ) )
= ( ~ ( member_a @ A3 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1154_insert__disjoint_I1_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1155_insert__Diff__single,axiom,
! [A3: a,A: set_a] :
( ( insert_a @ A3 @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( insert_a @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_1156_finite__Diff__insert,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1157_finite__Diff__insert,axiom,
! [A: set_a,A3: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1158_set__mset__replicate__mset__subset,axiom,
! [N2: nat,X: a] :
( ( ( N2 = zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N2 @ X ) )
= bot_bot_set_a ) )
& ( ( N2 != zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N2 @ X ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_1159_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A6: nat] :
( A3
= ( insert_nat @ A6 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1160_finite_Ocases,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A6: a] :
( A3
= ( insert_a @ A6 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1161_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A2: set_nat] :
( ( A2 = bot_bot_set_nat )
| ? [A4: set_nat,B4: nat] :
( ( A2
= ( insert_nat @ B4 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1162_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A2: set_a] :
( ( A2 = bot_bot_set_a )
| ? [A4: set_a,B4: a] :
( ( A2
= ( insert_a @ B4 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1163_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1164_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1165_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1166_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1167_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1168_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1169_subset__singleton__iff,axiom,
! [X5: set_a,A3: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1170_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1171_Diff__insert,axiom,
! [A: set_a,A3: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1172_insert__Diff,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ( ( insert_a @ A3 @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1173_Diff__insert2,axiom,
! [A: set_a,A3: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1174_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1175_insert__is__Un,axiom,
( insert_a
= ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1176_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1177_singleton__Un__iff,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1178_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C3 ) )
& ~ ( member_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1179_finite_OinsertI,axiom,
! [A: set_a,A3: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_1180_finite_OinsertI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_1181_insert__subsetI,axiom,
! [X: a,A: set_a,X5: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1182_singleton__inject,axiom,
! [A3: a,B2: a] :
( ( ( insert_a @ A3 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_1183_insert__not__empty,axiom,
! [A3: a,A: set_a] :
( ( insert_a @ A3 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1184_doubleton__eq__iff,axiom,
! [A3: a,B2: a,C2: a,D: a] :
( ( ( insert_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A3 = C2 )
& ( B2 = D ) )
| ( ( A3 = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1185_singleton__iff,axiom,
! [B2: a,A3: a] :
( ( member_a @ B2 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1186_singletonD,axiom,
! [B2: a,A3: a] :
( ( member_a @ B2 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1187_Int__insert__left,axiom,
! [A3: a,C3: set_a,B: set_a] :
( ( ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C3 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C3 ) ) ) )
& ( ~ ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C3 )
= ( inf_inf_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1188_Int__insert__right,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1189_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1190_insertE,axiom,
! [A3: a,B2: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B2 @ A ) )
=> ( ( A3 != B2 )
=> ( member_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_1191_insertI1,axiom,
! [A3: a,B: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B ) ) ).
% insertI1
thf(fact_1192_insertI2,axiom,
! [A3: a,B: set_a,B2: a] :
( ( member_a @ A3 @ B )
=> ( member_a @ A3 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1193_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1194_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1195_insert__absorb,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ( ( insert_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1196_insert__eq__iff,axiom,
! [A3: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A3 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A3 = B2 )
=> ( A = B ) )
& ( ( A3 != B2 )
=> ? [C5: set_a] :
( ( A
= ( insert_a @ B2 @ C5 ) )
& ~ ( member_a @ B2 @ C5 )
& ( B
= ( insert_a @ A3 @ C5 ) )
& ~ ( member_a @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1197_mk__disjoint__insert,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A3 @ B7 ) )
& ~ ( member_a @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1198_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1199_subset__insertI,axiom,
! [B: set_a,A3: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A3 @ B ) ) ).
% subset_insertI
thf(fact_1200_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1201_insert__mono,axiom,
! [C3: set_a,D2: set_a,A3: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A3 @ C3 ) @ ( insert_a @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_1202_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1203_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1204_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ B6 @ X6 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B6 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1205_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ X6 @ B6 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B6 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1206_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1207_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1208_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1209_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1210_infinite__remove,axiom,
! [S: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1211_infinite__remove,axiom,
! [S: set_a,A3: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1212_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A7: set_nat] :
( ( X5 @ A7 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A7 )
& ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1213_infinite__coinduct,axiom,
! [X5: set_a > $o,A: set_a] :
( ( X5 @ A )
=> ( ! [A7: set_a] :
( ( X5 @ A7 )
=> ? [X6: a] :
( ( member_a @ X6 @ A7 )
& ( ( X5 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1214_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A6: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A6 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1215_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A6: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A6 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1216_subset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1217_Diff__single__insert,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1218_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 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1219_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 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1220_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 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1221_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 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1222_card__Diff1__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ).
% card_Diff1_le
thf(fact_1223_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T4 ) )
& ( P @ ( insert_nat @ X6 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1224_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T4: set_a] :
( ( ord_less_set_a @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X6: a] :
( ( member_a @ X6 @ ( minus_minus_set_a @ S @ T4 ) )
& ( P @ ( insert_a @ X6 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1225_psubset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1226_card__Diff1__less,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_1227_card__Diff1__less,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_1228_card__Diff2__less,axiom,
! [A: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1229_card__Diff2__less,axiom,
! [A: set_a,X: a,Y: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1230_card__Diff1__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_1231_card__Diff1__less__iff,axiom,
! [A: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) )
= ( ( finite_finite_a @ A )
& ( member_a @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_1232_set__mset__single__iff__replicate__mset,axiom,
! [U: multiset_a,A3: a] :
( ( ( set_mset_a @ U )
= ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( U
= ( replicate_mset_a @ N4 @ A3 ) ) ) ) ) ).
% set_mset_single_iff_replicate_mset
thf(fact_1233_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_1234_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1235_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1236_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_1237_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X2: a] :
( A
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1238_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1239_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1240_add__0__iff,axiom,
! [B2: nat,A3: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1241_less__multiset_092_060_094sub_062D_092_060_094sub_062M,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M: multiset_nat,N: multiset_nat] :
? [X7: multiset_nat,Y6: multiset_nat] :
( ( X7 != zero_z7348594199698428585et_nat )
& ( subseteq_mset_nat @ X7 @ N )
& ( M
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ N @ X7 ) @ Y6 ) )
& ! [K3: nat] :
( ( member_nat @ K3 @ ( set_mset_nat @ Y6 ) )
=> ? [A2: nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ X7 ) )
& ( ord_less_nat @ K3 @ A2 ) ) ) ) ) ) ).
% less_multiset\<^sub>D\<^sub>M
thf(fact_1242_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1243_msubset__mset__set__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( subseteq_mset_nat @ ( mset_set_nat @ A ) @ ( mset_set_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_1244_msubset__mset__set__iff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( subseteq_mset_a @ ( mset_set_a @ A ) @ ( mset_set_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_1245_count__greater__eq__one__iff,axiom,
! [M2: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M2 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M2 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_1246_count__mset__set_I1_J,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( count_nat @ ( mset_set_nat @ A ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_1247_count__mset__set_I1_J,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( count_a @ ( mset_set_a @ A ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_1248_card__Diff__insert,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1249_subset__imp__msubset__mset__set,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( subseteq_mset_nat @ ( mset_set_nat @ A ) @ ( mset_set_nat @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_1250_subset__imp__msubset__mset__set,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( subseteq_mset_a @ ( mset_set_a @ A ) @ ( mset_set_a @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_1251_mset__subset__eqD,axiom,
! [A: multiset_a,B: multiset_a,X: a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( member_a @ X @ ( set_mset_a @ A ) )
=> ( member_a @ X @ ( set_mset_a @ B ) ) ) ) ).
% mset_subset_eqD
thf(fact_1252_mset__set__set__mset__subseteq,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ ( mset_set_a @ ( set_mset_a @ A ) ) @ A ) ).
% mset_set_set_mset_subseteq
thf(fact_1253_set__mset__mono,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A ) @ ( set_mset_a @ B ) ) ) ).
% set_mset_mono
thf(fact_1254_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1255_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1256_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1257_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1258_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1259_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1260_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1261_mset__subset__eqI,axiom,
! [A: multiset_a,B: multiset_a] :
( ! [A6: a] : ( ord_less_eq_nat @ ( count_a @ A @ A6 ) @ ( count_a @ B @ A6 ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% mset_subset_eqI
thf(fact_1262_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A4: multiset_a,B3: multiset_a] :
! [A2: a] : ( ord_less_eq_nat @ ( count_a @ A4 @ A2 ) @ ( count_a @ B3 @ A2 ) ) ) ) ).
% subseteq_mset_def
thf(fact_1263_mset__subset__eq__count,axiom,
! [A: multiset_a,B: multiset_a,A3: a] :
( ( subseteq_mset_a @ A @ B )
=> ( ord_less_eq_nat @ ( count_a @ A @ A3 ) @ ( count_a @ B @ A3 ) ) ) ).
% mset_subset_eq_count
thf(fact_1264_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
% Conjectures (2)
thf(conj_0,hypothesis,
member_a @ x @ c ).
thf(conj_1,conjecture,
( ( count_a @ a2 @ x )
= zero_zero_nat ) ).
%------------------------------------------------------------------------------