TPTP Problem File: SLH0566^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0022_Restrict_Bounds/prob_00090_003355__17629942_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1523 ( 624 unt; 231 typ; 0 def)
% Number of atoms : 4181 (1609 equ; 0 cnn)
% Maximal formula atoms : 31 ( 3 avg)
% Number of connectives : 11958 ( 565 ~; 90 |; 425 &;9066 @)
% ( 0 <=>;1812 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 1441 (1441 >; 0 *; 0 +; 0 <<)
% Number of symbols : 215 ( 212 usr; 16 con; 0-4 aty)
% Number of variables : 4036 ( 615 ^;3226 !; 195 ?;4036 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:45.206
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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% Explicit typings (212)
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image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
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image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert6993385188428902846la_a_b: ( nat > relational_fmla_a_b ) > set_na7556516505497143492la_a_b > set_na7556516505497143492la_a_b ).
thf(sy_c_Set_Oinsert_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
insert2915755147076878910_b_nat: ( relational_fmla_a_b > nat ) > set_Re5563258083572488516_b_nat > set_Re5563258083572488516_b_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert8904949763332019597la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re1288005135514575379la_a_b > set_Re1288005135514575379la_a_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert2023870700798818565la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_emp6953259385542938189la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
is_sin1114528679824004833la_a_b: set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
remove4261432235257513082la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
remove6344774115224635824la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
accp_R989495437599811158la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Oterm_It__Nat__Onat_J,type,
accp_R1321120385092908946rm_nat: ( relational_term_nat > relational_term_nat > $o ) > relational_term_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Oterm_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
accp_R2749021782369031905la_a_b: ( relati112041753218324778la_a_b > relati112041753218324778la_a_b > $o ) > relati112041753218324778la_a_b > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member8923333377441230501la_a_b: ( nat > relational_fmla_a_b ) > set_na7556516505497143492la_a_b > $o ).
thf(sy_c_member_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
member4845703336089206565_b_nat: ( relational_fmla_a_b > nat ) > set_Re5563258083572488516_b_nat > $o ).
thf(sy_c_member_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member8433577210552456052la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re1288005135514575379la_a_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v__092_060Q_062,type,
q2: set_Re381260168593705685la_a_b ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (1275)
thf(fact_0_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_1_empty__iff,axiom,
! [C: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ C @ bot_bo2891247006866115487la_a_b ) ).
% empty_iff
thf(fact_2_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_3_empty__iff,axiom,
! [C: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_4_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_5_all__not__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ! [X: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X @ A ) )
= ( A = bot_bo2891247006866115487la_a_b ) ) ).
% all_not_in_conv
thf(fact_6_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_7_all__not__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ! [X: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_8_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_9_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_10_Collect__empty__eq,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o] :
( ( ( collec4193374254315349731la_a_b @ P )
= bot_bo5171125281866047832la_a_b )
= ( ! [X: nat > relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_11_Collect__empty__eq,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o] :
( ( ( collec115744212963325795_b_nat @ P )
= bot_bo3177866859941392856_b_nat )
= ( ! [X: relational_fmla_a_b > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_12_Collect__empty__eq,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ( collec5041345499257167282la_a_b @ P )
= bot_bo9179849999556691623la_a_b )
= ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_13_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_14_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X: relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_15_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_16_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_17_empty__Collect__eq,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o] :
( ( bot_bo5171125281866047832la_a_b
= ( collec4193374254315349731la_a_b @ P ) )
= ( ! [X: nat > relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o] :
( ( bot_bo3177866859941392856_b_nat
= ( collec115744212963325795_b_nat @ P ) )
= ( ! [X: relational_fmla_a_b > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( bot_bo9179849999556691623la_a_b
= ( collec5041345499257167282la_a_b @ P ) )
= ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X: relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_22_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_23_bot__apply,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X: relational_fmla_a_b] : bot_bot_o ) ) ).
% bot_apply
thf(fact_24_Set_Oempty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X: set_nat] : $false ) ) ).
% Set.empty_def
thf(fact_25_Set_Oempty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X: nat > nat] : $false ) ) ).
% Set.empty_def
thf(fact_26_Set_Oempty__def,axiom,
( bot_bo5171125281866047832la_a_b
= ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] : $false ) ) ).
% Set.empty_def
thf(fact_27_Set_Oempty__def,axiom,
( bot_bo3177866859941392856_b_nat
= ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] : $false ) ) ).
% Set.empty_def
thf(fact_28_Set_Oempty__def,axiom,
( bot_bo9179849999556691623la_a_b
= ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : $false ) ) ).
% Set.empty_def
thf(fact_29_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_30_Set_Oempty__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : $false ) ) ).
% Set.empty_def
thf(fact_31_nongens__def,axiom,
( relati62690040636126068ns_a_b
= ( ^ [Q: relational_fmla_a_b] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ~ ? [X2: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X2 ) ) ) ) ) ).
% nongens_def
thf(fact_32_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_33_emptyE,axiom,
! [A2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ).
% emptyE
thf(fact_34_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_35_emptyE,axiom,
! [A2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_36_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_37_equals0D,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( A = bot_bo2891247006866115487la_a_b )
=> ~ ( member3481406638322139244la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_38_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_39_equals0D,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( A = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_40_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_41_equals0I,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ! [Y: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ Y @ A )
=> ( A = bot_bo2891247006866115487la_a_b ) ) ).
% equals0I
thf(fact_42_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_43_equals0I,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [Y: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y @ A )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_44_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_45_ex__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ? [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A ) )
= ( A != bot_bo2891247006866115487la_a_b ) ) ).
% ex_in_conv
thf(fact_46_ex__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ? [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A ) )
= ( A != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_47_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_48_set__notEmptyE,axiom,
! [S: set_set_nat] :
( ( S != bot_bot_set_set_nat )
=> ~ ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_49_set__notEmptyE,axiom,
! [S: set_se6865892389300016395la_a_b] :
( ( S != bot_bo2891247006866115487la_a_b )
=> ~ ! [X3: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_50_set__notEmptyE,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_51_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_52_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_53_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_54_bot__set__def,axiom,
( bot_bo5171125281866047832la_a_b
= ( collec4193374254315349731la_a_b @ bot_bo1911532952769645357_a_b_o ) ) ).
% bot_set_def
thf(fact_55_bot__set__def,axiom,
( bot_bo3177866859941392856_b_nat
= ( collec115744212963325795_b_nat @ bot_bo8935079179648391853_nat_o ) ) ).
% bot_set_def
thf(fact_56_bot__set__def,axiom,
( bot_bo9179849999556691623la_a_b
= ( collec5041345499257167282la_a_b @ bot_bo845822638115665054_a_b_o ) ) ).
% bot_set_def
thf(fact_57_bot__set__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ bot_bo8852203127187332700_a_b_o ) ) ).
% bot_set_def
thf(fact_58_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_59_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_60_bot__fun__def,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X: relational_fmla_a_b] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_61_memb__imp__not__empty,axiom,
! [X4: set_nat,S: set_set_nat] :
( ( member_set_nat @ X4 @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_62_memb__imp__not__empty,axiom,
! [X4: set_Re381260168593705685la_a_b,S: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ S )
=> ( S != bot_bo2891247006866115487la_a_b ) ) ).
% memb_imp_not_empty
thf(fact_63_memb__imp__not__empty,axiom,
! [X4: relational_fmla_a_b,S: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% memb_imp_not_empty
thf(fact_64_memb__imp__not__empty,axiom,
! [X4: nat,S: set_nat] :
( ( member_nat @ X4 @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_65_Set_Ois__empty__def,axiom,
( is_emp6953259385542938189la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] : ( A3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_66_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_67_fixbound__def,axiom,
( restri6210327484173102631nd_a_b
= ( ^ [Q2: set_Re381260168593705685la_a_b,X: nat] :
( collec3419995626248312948la_a_b
@ ^ [Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q @ Q2 )
& ( member_nat @ X @ ( relati62690040636126068ns_a_b @ Q ) ) ) ) ) ) ).
% fixbound_def
thf(fact_68_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_69_Collect__empty__eq__bot,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( P = bot_bot_nat_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_70_Collect__empty__eq__bot,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o] :
( ( ( collec4193374254315349731la_a_b @ P )
= bot_bo5171125281866047832la_a_b )
= ( P = bot_bo1911532952769645357_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_71_Collect__empty__eq__bot,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o] :
( ( ( collec115744212963325795_b_nat @ P )
= bot_bo3177866859941392856_b_nat )
= ( P = bot_bo8935079179648391853_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_72_Collect__empty__eq__bot,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ( collec5041345499257167282la_a_b @ P )
= bot_bo9179849999556691623la_a_b )
= ( P = bot_bo845822638115665054_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_73_Collect__empty__eq__bot,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( P = bot_bo8852203127187332700_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_74_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_75_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_76_bot__empty__eq,axiom,
( bot_bo8397419132826146726_a_b_o
= ( ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_77_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_78_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_79_gen__Bool__False,axiom,
! [X4: nat,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ ( relational_Bool_a_b @ $false ) @ G )
= ( G = bot_bo4495933725496725865la_a_b ) ) ).
% gen_Bool_False
thf(fact_80_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( member_set_nat @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_81_is__singletonI_H,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( A != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( ( member3481406638322139244la_a_b @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin1114528679824004833la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_82_is__singletonI_H,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( ( member4680049679412964150la_a_b @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin6594375743535830443la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_83_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_84_qp__Gen,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_qp_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ Q3 @ X_1 ) ) ) ).
% qp_Gen
thf(fact_85_gen__fv,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q3 ) ) ) ) ) ).
% gen_fv
thf(fact_86_gen__genempty,axiom,
! [Z: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ Q3 @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( relati5999705594545617851ty_a_b @ Q3 ) ) ) ).
% gen_genempty
thf(fact_87_cov_H_Ononfree,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_cov_a_b2 @ X4 @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ).
% cov'.nonfree
thf(fact_88_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_89_dual__order_Orefl,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_90_dual__order_Orefl,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_91_dual__order_Orefl,axiom,
! [A2: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_92_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_93_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_94_subsetI,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( member3481406638322139244la_a_b @ X3 @ B ) )
=> ( ord_le1577343677690852715la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_95_subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( member4680049679412964150la_a_b @ X3 @ B ) )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_96_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_97_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_98_subset__antisym,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_99_fmla_Oinject_I2_J,axiom,
! [X22: $o,Y2: $o] :
( ( ( relational_Bool_a_b @ X22 )
= ( relational_Bool_a_b @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fmla.inject(2)
thf(fact_100_subset__empty,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% subset_empty
thf(fact_101_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_102_empty__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A ) ).
% empty_subsetI
thf(fact_103_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_104_gen__Bool__True,axiom,
! [X4: nat,G: set_Re381260168593705685la_a_b] :
~ ( relational_gen_a_b @ X4 @ ( relational_Bool_a_b @ $true ) @ G ) ).
% gen_Bool_True
thf(fact_105_genempty_Ointros_I1_J,axiom,
relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).
% genempty.intros(1)
thf(fact_106_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_107_le__cases3,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_108_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_109_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( Y4 = Z2 ) )
= ( ^ [X: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ Y5 )
& ( ord_le4112832032246704949la_a_b @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_110_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
= ( ^ [X: nat > $o,Y5: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y5 )
& ( ord_less_eq_nat_o @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_111_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: relational_fmla_a_b > $o,Z2: relational_fmla_a_b > $o] : ( Y4 = Z2 ) )
= ( ^ [X: relational_fmla_a_b > $o,Y5: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ X @ Y5 )
& ( ord_le7191224889845164944_a_b_o @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_112_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_113_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_114_ord__eq__le__trans,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( A2 = B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C )
=> ( ord_le4112832032246704949la_a_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_115_ord__eq__le__trans,axiom,
! [A2: nat > $o,B2: nat > $o,C: nat > $o] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ord_less_eq_nat_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_116_ord__eq__le__trans,axiom,
! [A2: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( A2 = B2 )
=> ( ( ord_le7191224889845164944_a_b_o @ B2 @ C )
=> ( ord_le7191224889845164944_a_b_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_117_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_118_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_119_ord__le__eq__trans,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le4112832032246704949la_a_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_120_ord__le__eq__trans,axiom,
! [A2: nat > $o,B2: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_121_ord__le__eq__trans,axiom,
! [A2: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le7191224889845164944_a_b_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_122_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_123_order__antisym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_124_order__antisym,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X4 @ Y3 )
=> ( ( ord_le4112832032246704949la_a_b @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_125_order__antisym,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ( ord_less_eq_nat_o @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_126_order__antisym,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 )
=> ( ( ord_le7191224889845164944_a_b_o @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_127_order__antisym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_128_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_129_order_Otrans,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C )
=> ( ord_le4112832032246704949la_a_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_130_order_Otrans,axiom,
! [A2: nat > $o,B2: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ord_less_eq_nat_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_131_order_Otrans,axiom,
! [A2: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ B2 )
=> ( ( ord_le7191224889845164944_a_b_o @ B2 @ C )
=> ( ord_le7191224889845164944_a_b_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_132_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_133_order__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z )
=> ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X4 @ Y3 )
=> ( ( ord_le4112832032246704949la_a_b @ Y3 @ Z )
=> ( ord_le4112832032246704949la_a_b @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_135_order__trans,axiom,
! [X4: nat > $o,Y3: nat > $o,Z: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ( ord_less_eq_nat_o @ Y3 @ Z )
=> ( ord_less_eq_nat_o @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_136_order__trans,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 )
=> ( ( ord_le7191224889845164944_a_b_o @ Y3 @ Z )
=> ( ord_le7191224889845164944_a_b_o @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_137_order__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_138_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_139_mem__Collect__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ A2 @ ( collec2099942116761351594la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_140_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_141_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_142_mem__Collect__eq,axiom,
! [A2: nat > relational_fmla_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ( member8923333377441230501la_a_b @ A2 @ ( collec4193374254315349731la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_143_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b > nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ( member4845703336089206565_b_nat @ A2 @ ( collec115744212963325795_b_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b > relational_fmla_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( member8433577210552456052la_a_b @ A2 @ ( collec5041345499257167282la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_na7556516505497143492la_a_b] :
( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] : ( member8923333377441230501la_a_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_Re5563258083572488516_b_nat] :
( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] : ( member4845703336089206565_b_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
! [A: set_Re1288005135514575379la_a_b] :
( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_155_Collect__cong,axiom,
! [P: set_nat > $o,Q3: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_156_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q3: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_157_Collect__cong,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o,Q3: ( nat > relational_fmla_a_b ) > $o] :
( ! [X3: nat > relational_fmla_a_b] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec4193374254315349731la_a_b @ P )
= ( collec4193374254315349731la_a_b @ Q3 ) ) ) ).
% Collect_cong
thf(fact_158_Collect__cong,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o,Q3: ( relational_fmla_a_b > nat ) > $o] :
( ! [X3: relational_fmla_a_b > nat] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec115744212963325795_b_nat @ P )
= ( collec115744212963325795_b_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_159_Collect__cong,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,Q3: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ! [X3: relational_fmla_a_b > relational_fmla_a_b] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec5041345499257167282la_a_b @ P )
= ( collec5041345499257167282la_a_b @ Q3 ) ) ) ).
% Collect_cong
thf(fact_160_Collect__cong,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec3419995626248312948la_a_b @ P )
= ( collec3419995626248312948la_a_b @ Q3 ) ) ) ).
% Collect_cong
thf(fact_161_Collect__cong,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_162_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_163_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B4 @ A5 )
& ( ord_le4112832032246704949la_a_b @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A5 )
& ( ord_less_eq_nat_o @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: relational_fmla_a_b > $o,Z2: relational_fmla_a_b > $o] : ( Y4 = Z2 ) )
= ( ^ [A5: relational_fmla_a_b > $o,B4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B4 @ A5 )
& ( ord_le7191224889845164944_a_b_o @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_167_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_168_dual__order_Oantisym,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_169_dual__order_Oantisym,axiom,
! [B2: nat > $o,A2: nat > $o] :
( ( ord_less_eq_nat_o @ B2 @ A2 )
=> ( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_170_dual__order_Oantisym,axiom,
! [B2: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B2 @ A2 )
=> ( ( ord_le7191224889845164944_a_b_o @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_171_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_172_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_173_dual__order_Otrans,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ C @ B2 )
=> ( ord_le4112832032246704949la_a_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_174_dual__order_Otrans,axiom,
! [B2: nat > $o,A2: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B2 @ A2 )
=> ( ( ord_less_eq_nat_o @ C @ B2 )
=> ( ord_less_eq_nat_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_175_dual__order_Otrans,axiom,
! [B2: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B2 @ A2 )
=> ( ( ord_le7191224889845164944_a_b_o @ C @ B2 )
=> ( ord_le7191224889845164944_a_b_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_176_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_177_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_178_in__mono,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ X4 @ A )
=> ( member3481406638322139244la_a_b @ X4 @ B ) ) ) ).
% in_mono
thf(fact_179_in__mono,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( member4680049679412964150la_a_b @ X4 @ B ) ) ) ).
% in_mono
thf(fact_180_in__mono,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_181_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_182_subsetD,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_183_subsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_184_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_185_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_186_equalityE,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A = B )
=> ~ ( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ~ ( ord_le4112832032246704949la_a_b @ B @ A ) ) ) ).
% equalityE
thf(fact_187_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( member_set_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_188_subset__eq,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
! [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A3 )
=> ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_189_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A3 )
=> ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_190_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_191_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_192_equalityD1,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A = B )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ).
% equalityD1
thf(fact_193_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_194_Set_OequalityD2,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A = B )
=> ( ord_le4112832032246704949la_a_b @ B @ A ) ) ).
% Set.equalityD2
thf(fact_195_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_196_subset__iff,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
! [T: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ T @ A3 )
=> ( member3481406638322139244la_a_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_197_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
! [T: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T @ A3 )
=> ( member4680049679412964150la_a_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_198_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_199_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_200_subset__refl,axiom,
! [A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A @ A ) ).
% subset_refl
thf(fact_201_Collect__mono,axiom,
! [P: set_nat > $o,Q3: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_202_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q3: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_203_Collect__mono,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o,Q3: ( nat > relational_fmla_a_b ) > $o] :
( ! [X3: nat > relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le3065308906375977252la_a_b @ ( collec4193374254315349731la_a_b @ P ) @ ( collec4193374254315349731la_a_b @ Q3 ) ) ) ).
% Collect_mono
thf(fact_204_Collect__mono,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o,Q3: ( relational_fmla_a_b > nat ) > $o] :
( ! [X3: relational_fmla_a_b > nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le1072050484451322276_b_nat @ ( collec115744212963325795_b_nat @ P ) @ ( collec115744212963325795_b_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_205_Collect__mono,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,Q3: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ! [X3: relational_fmla_a_b > relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le116059350455505523la_a_b @ ( collec5041345499257167282la_a_b @ P ) @ ( collec5041345499257167282la_a_b @ Q3 ) ) ) ).
% Collect_mono
thf(fact_206_Collect__mono,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le4112832032246704949la_a_b @ ( collec3419995626248312948la_a_b @ P ) @ ( collec3419995626248312948la_a_b @ Q3 ) ) ) ).
% Collect_mono
thf(fact_207_Collect__mono,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_208_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_209_subset__trans,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C2 )
=> ( ord_le4112832032246704949la_a_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_210_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_211_antisym,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_212_antisym,axiom,
! [A2: nat > $o,B2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_213_antisym,axiom,
! [A2: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ B2 )
=> ( ( ord_le7191224889845164944_a_b_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_214_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_215_le__funD,axiom,
! [F: nat > $o,G2: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funD
thf(fact_216_le__funD,axiom,
! [F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o,X4: relational_fmla_a_b] :
( ( ord_le7191224889845164944_a_b_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funD
thf(fact_217_le__funE,axiom,
! [F: nat > $o,G2: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funE
thf(fact_218_le__funE,axiom,
! [F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o,X4: relational_fmla_a_b] :
( ( ord_le7191224889845164944_a_b_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funE
thf(fact_219_le__funI,axiom,
! [F: nat > $o,G2: nat > $o] :
( ! [X3: nat] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_220_le__funI,axiom,
! [F: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le7191224889845164944_a_b_o @ F @ G2 ) ) ).
% le_funI
thf(fact_221_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_222_set__eq__subset,axiom,
( ( ^ [Y4: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( Y4 = Z2 ) )
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A3 @ B5 )
& ( ord_le4112832032246704949la_a_b @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_223_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G3: nat > $o] :
! [X: nat] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_fun_def
thf(fact_224_le__fun__def,axiom,
( ord_le7191224889845164944_a_b_o
= ( ^ [F2: relational_fmla_a_b > $o,G3: relational_fmla_a_b > $o] :
! [X: relational_fmla_a_b] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_fun_def
thf(fact_225_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q3: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q3 ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_226_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q3: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q3 ) )
= ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_227_Collect__mono__iff,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o,Q3: ( nat > relational_fmla_a_b ) > $o] :
( ( ord_le3065308906375977252la_a_b @ ( collec4193374254315349731la_a_b @ P ) @ ( collec4193374254315349731la_a_b @ Q3 ) )
= ( ! [X: nat > relational_fmla_a_b] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_228_Collect__mono__iff,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o,Q3: ( relational_fmla_a_b > nat ) > $o] :
( ( ord_le1072050484451322276_b_nat @ ( collec115744212963325795_b_nat @ P ) @ ( collec115744212963325795_b_nat @ Q3 ) )
= ( ! [X: relational_fmla_a_b > nat] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_229_Collect__mono__iff,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,Q3: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ord_le116059350455505523la_a_b @ ( collec5041345499257167282la_a_b @ P ) @ ( collec5041345499257167282la_a_b @ Q3 ) )
= ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_230_Collect__mono__iff,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ ( collec3419995626248312948la_a_b @ P ) @ ( collec3419995626248312948la_a_b @ Q3 ) )
= ( ! [X: relational_fmla_a_b] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_231_Collect__mono__iff,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q3 ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q3 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_232_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_233_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A5 @ B4 )
& ( ord_le4112832032246704949la_a_b @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_234_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A5: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A5 @ B4 )
& ( ord_less_eq_nat_o @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_235_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: relational_fmla_a_b > $o,Z2: relational_fmla_a_b > $o] : ( Y4 = Z2 ) )
= ( ^ [A5: relational_fmla_a_b > $o,B4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A5 @ B4 )
& ( ord_le7191224889845164944_a_b_o @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_236_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_237_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_238_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_239_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_240_order__subst1,axiom,
! [A2: nat,F: ( nat > $o ) > nat,B2: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_241_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst1,axiom,
! [A2: nat > $o,F: nat > nat > $o,B2: nat,C: nat] :
( ( ord_less_eq_nat_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_243_order__subst1,axiom,
! [A2: nat,F: set_Re381260168593705685la_a_b > nat,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_244_order__subst1,axiom,
! [A2: set_nat,F: ( nat > $o ) > set_nat,B2: nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_245_order__subst1,axiom,
! [A2: set_Re381260168593705685la_a_b,F: nat > set_Re381260168593705685la_a_b,B2: nat,C: nat] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4112832032246704949la_a_b @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_246_order__subst1,axiom,
! [A2: nat > $o,F: set_nat > nat > $o,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_247_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_248_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_249_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_250_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_o @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_251_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_252_order__subst2,axiom,
! [A2: nat > $o,B2: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_253_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le4112832032246704949la_a_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4112832032246704949la_a_b @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4112832032246704949la_a_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_254_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_o @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_255_order__subst2,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > nat,C: nat] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_256_order__subst2,axiom,
! [A2: nat > $o,B2: nat > $o,F: ( nat > $o ) > set_nat,C: set_nat] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_257_ord__eq__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( C = D )
=> ( ord_less_eq_set_nat @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_258_ord__eq__le__eq__trans,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( A2 = B2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C )
=> ( ( C = D )
=> ( ord_le4112832032246704949la_a_b @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_259_ord__eq__le__eq__trans,axiom,
! [A2: nat > $o,B2: nat > $o,C: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ( C = D )
=> ( ord_less_eq_nat_o @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_260_ord__eq__le__eq__trans,axiom,
! [A2: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o,D: relational_fmla_a_b > $o] :
( ( A2 = B2 )
=> ( ( ord_le7191224889845164944_a_b_o @ B2 @ C )
=> ( ( C = D )
=> ( ord_le7191224889845164944_a_b_o @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_261_ord__eq__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( C = D )
=> ( ord_less_eq_nat @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_262_order__eq__refl,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_263_order__eq__refl,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( X4 = Y3 )
=> ( ord_le4112832032246704949la_a_b @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_264_order__eq__refl,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat_o @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_265_order__eq__refl,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( X4 = Y3 )
=> ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_266_order__eq__refl,axiom,
! [X4: nat,Y3: nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_267_subset__Collect__conv,axiom,
! [S: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ S @ ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_268_subset__Collect__conv,axiom,
! [S: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ S @ ( collect_nat_nat @ P ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_269_subset__Collect__conv,axiom,
! [S: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ( ord_le3065308906375977252la_a_b @ S @ ( collec4193374254315349731la_a_b @ P ) )
= ( ! [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_270_subset__Collect__conv,axiom,
! [S: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ( ord_le1072050484451322276_b_nat @ S @ ( collec115744212963325795_b_nat @ P ) )
= ( ! [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_271_subset__Collect__conv,axiom,
! [S: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ord_le116059350455505523la_a_b @ S @ ( collec5041345499257167282la_a_b @ P ) )
= ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_272_subset__Collect__conv,axiom,
! [S: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ S @ ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_273_subset__Collect__conv,axiom,
! [S: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
= ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_274_linorder__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_linear
thf(fact_275_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_276_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_277_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_278_ord__eq__le__subst,axiom,
! [A2: nat > $o,F: nat > nat > $o,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_279_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
! [A2: nat,F: ( nat > $o ) > nat,B2: nat > $o,C: nat > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_281_ord__eq__le__subst,axiom,
! [A2: set_Re381260168593705685la_a_b,F: nat > set_Re381260168593705685la_a_b,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4112832032246704949la_a_b @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_282_ord__eq__le__subst,axiom,
! [A2: nat > $o,F: set_nat > nat > $o,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_283_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Re381260168593705685la_a_b > nat,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ C )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_284_ord__eq__le__subst,axiom,
! [A2: set_nat,F: ( nat > $o ) > set_nat,B2: nat > $o,C: nat > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_o @ B2 @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_285_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_286_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_287_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_288_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_289_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_290_ord__le__eq__subst,axiom,
! [A2: nat > $o,B2: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4112832032246704949la_a_b @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4112832032246704949la_a_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_ord__le__eq__subst,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > nat,C: nat] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_294_ord__le__eq__subst,axiom,
! [A2: nat > $o,B2: nat > $o,F: ( nat > $o ) > set_nat,C: set_nat] :
( ( ord_less_eq_nat_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_295_linorder__le__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_le_cases
thf(fact_296_order__antisym__conv,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_297_order__antisym__conv,axiom,
! [Y3: set_Re381260168593705685la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y3 @ X4 )
=> ( ( ord_le4112832032246704949la_a_b @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_298_order__antisym__conv,axiom,
! [Y3: nat > $o,X4: nat > $o] :
( ( ord_less_eq_nat_o @ Y3 @ X4 )
=> ( ( ord_less_eq_nat_o @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_299_order__antisym__conv,axiom,
! [Y3: relational_fmla_a_b > $o,X4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ Y3 @ X4 )
=> ( ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_300_order__antisym__conv,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_301_fixbound__in,axiom,
! [Q3: relational_fmla_a_b,Q4: set_Re381260168593705685la_a_b,X4: nat] :
( ( member4680049679412964150la_a_b @ Q3 @ ( restri6210327484173102631nd_a_b @ Q4 @ X4 ) )
=> ( member4680049679412964150la_a_b @ Q3 @ Q4 ) ) ).
% fixbound_in
thf(fact_302_Collect__subset,axiom,
! [A: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ord_le1577343677690852715la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_303_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_304_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_305_Collect__subset,axiom,
! [A: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ord_le3065308906375977252la_a_b
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_306_Collect__subset,axiom,
! [A: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ord_le1072050484451322276_b_nat
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_307_Collect__subset,axiom,
! [A: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ord_le116059350455505523la_a_b
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_308_Collect__subset,axiom,
! [A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_309_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_310_fv_Osimps_I2_J,axiom,
! [B2: $o] :
( ( relational_fv_a_b @ ( relational_Bool_a_b @ B2 ) )
= bot_bot_set_nat ) ).
% fv.simps(2)
thf(fact_311_bot_Oextremum,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A2 ) ).
% bot.extremum
thf(fact_312_bot_Oextremum,axiom,
! [A2: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ bot_bo8852203127187332700_a_b_o @ A2 ) ).
% bot.extremum
thf(fact_313_bot_Oextremum,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ).
% bot.extremum
thf(fact_314_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_315_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_316_bot_Oextremum__unique,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ bot_bot_nat_o )
= ( A2 = bot_bot_nat_o ) ) ).
% bot.extremum_unique
thf(fact_317_bot_Oextremum__unique,axiom,
! [A2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ bot_bo8852203127187332700_a_b_o )
= ( A2 = bot_bo8852203127187332700_a_b_o ) ) ).
% bot.extremum_unique
thf(fact_318_bot_Oextremum__unique,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_unique
thf(fact_319_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_320_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_321_bot_Oextremum__uniqueI,axiom,
! [A2: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ bot_bot_nat_o )
=> ( A2 = bot_bot_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_322_bot_Oextremum__uniqueI,axiom,
! [A2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ bot_bo8852203127187332700_a_b_o )
=> ( A2 = bot_bo8852203127187332700_a_b_o ) ) ).
% bot.extremum_uniqueI
thf(fact_323_bot_Oextremum__uniqueI,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
=> ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_324_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_325_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_326_gen__qp,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen_qp
thf(fact_327_fixbound__fv,axiom,
! [Q3: relational_fmla_a_b,Q4: set_Re381260168593705685la_a_b,X4: nat] :
( ( member4680049679412964150la_a_b @ Q3 @ ( restri6210327484173102631nd_a_b @ Q4 @ X4 ) )
=> ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) ) ) ).
% fixbound_fv
thf(fact_328_gen_Ointros_I1_J,axiom,
! [X4: nat] : ( relational_gen_a_b @ X4 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen.intros(1)
thf(fact_329_order__mono__setup_Orefl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_330_order__mono__setup_Orefl,axiom,
! [X4: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_331_order__mono__setup_Orefl,axiom,
! [X4: nat > $o] : ( ord_less_eq_nat_o @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_332_order__mono__setup_Orefl,axiom,
! [X4: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_333_order__mono__setup_Orefl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_334_subset__emptyI,axiom,
! [A: set_set_nat] :
( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_335_subset__emptyI,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X3 @ A )
=> ( ord_le1577343677690852715la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ).
% subset_emptyI
thf(fact_336_subset__emptyI,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% subset_emptyI
thf(fact_337_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_338_subset__Collect__iff,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ord_le1577343677690852715la_a_b @ B @ A )
=> ( ( ord_le1577343677690852715la_a_b @ B
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_339_subset__Collect__iff,axiom,
! [B: set_set_nat,A: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ B
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_340_subset__Collect__iff,axiom,
! [B: set_nat_nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ B
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_341_subset__Collect__iff,axiom,
! [B: set_na7556516505497143492la_a_b,A: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ( ord_le3065308906375977252la_a_b @ B @ A )
=> ( ( ord_le3065308906375977252la_a_b @ B
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_342_subset__Collect__iff,axiom,
! [B: set_Re5563258083572488516_b_nat,A: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ( ord_le1072050484451322276_b_nat @ B @ A )
=> ( ( ord_le1072050484451322276_b_nat @ B
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_343_subset__Collect__iff,axiom,
! [B: set_Re1288005135514575379la_a_b,A: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ord_le116059350455505523la_a_b @ B @ A )
=> ( ( ord_le116059350455505523la_a_b @ B
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_344_subset__Collect__iff,axiom,
! [B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ B @ A )
=> ( ( ord_le4112832032246704949la_a_b @ B
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_345_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_346_subset__CollectI,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b,Q3: set_Re381260168593705685la_a_b > $o,P: set_Re381260168593705685la_a_b > $o] :
( ( ord_le1577343677690852715la_a_b @ B @ A )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1577343677690852715la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ B )
& ( Q3 @ X ) ) )
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_347_subset__CollectI,axiom,
! [B: set_set_nat,A: set_set_nat,Q3: set_nat > $o,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ B )
& ( Q3 @ X ) ) )
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_348_subset__CollectI,axiom,
! [B: set_nat_nat,A: set_nat_nat,Q3: ( nat > nat ) > $o,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ B )
& ( Q3 @ X ) ) )
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_349_subset__CollectI,axiom,
! [B: set_na7556516505497143492la_a_b,A: set_na7556516505497143492la_a_b,Q3: ( nat > relational_fmla_a_b ) > $o,P: ( nat > relational_fmla_a_b ) > $o] :
( ( ord_le3065308906375977252la_a_b @ B @ A )
=> ( ! [X3: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le3065308906375977252la_a_b
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ B )
& ( Q3 @ X ) ) )
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_350_subset__CollectI,axiom,
! [B: set_Re5563258083572488516_b_nat,A: set_Re5563258083572488516_b_nat,Q3: ( relational_fmla_a_b > nat ) > $o,P: ( relational_fmla_a_b > nat ) > $o] :
( ( ord_le1072050484451322276_b_nat @ B @ A )
=> ( ! [X3: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1072050484451322276_b_nat
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ B )
& ( Q3 @ X ) ) )
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_351_subset__CollectI,axiom,
! [B: set_Re1288005135514575379la_a_b,A: set_Re1288005135514575379la_a_b,Q3: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ord_le116059350455505523la_a_b @ B @ A )
=> ( ! [X3: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le116059350455505523la_a_b
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ B )
& ( Q3 @ X ) ) )
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_352_subset__CollectI,axiom,
! [B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,Q3: relational_fmla_a_b > $o,P: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ B @ A )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
& ( Q3 @ X ) ) )
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_353_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q3: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q3 @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_354_Collect__restrict,axiom,
! [X5: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ord_le1577343677690852715la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_355_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_356_Collect__restrict,axiom,
! [X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_357_Collect__restrict,axiom,
! [X5: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ord_le3065308906375977252la_a_b
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_358_Collect__restrict,axiom,
! [X5: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ord_le1072050484451322276_b_nat
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_359_Collect__restrict,axiom,
! [X5: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ord_le116059350455505523la_a_b
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_360_Collect__restrict,axiom,
! [X5: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_361_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_362_prop__restrict,axiom,
! [X4: set_Re381260168593705685la_a_b,Z3: set_se6865892389300016395la_a_b,X5: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ X4 @ Z3 )
=> ( ( ord_le1577343677690852715la_a_b @ Z3
@ ( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_363_prop__restrict,axiom,
! [X4: set_nat,Z3: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X4 @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_364_prop__restrict,axiom,
! [X4: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X4 @ Z3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_365_prop__restrict,axiom,
! [X4: nat > relational_fmla_a_b,Z3: set_na7556516505497143492la_a_b,X5: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ( member8923333377441230501la_a_b @ X4 @ Z3 )
=> ( ( ord_le3065308906375977252la_a_b @ Z3
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_366_prop__restrict,axiom,
! [X4: relational_fmla_a_b > nat,Z3: set_Re5563258083572488516_b_nat,X5: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ( member4845703336089206565_b_nat @ X4 @ Z3 )
=> ( ( ord_le1072050484451322276_b_nat @ Z3
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_367_prop__restrict,axiom,
! [X4: relational_fmla_a_b > relational_fmla_a_b,Z3: set_Re1288005135514575379la_a_b,X5: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( member8433577210552456052la_a_b @ X4 @ Z3 )
=> ( ( ord_le116059350455505523la_a_b @ Z3
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_368_prop__restrict,axiom,
! [X4: relational_fmla_a_b,Z3: set_Re381260168593705685la_a_b,X5: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X4 @ Z3 )
=> ( ( ord_le4112832032246704949la_a_b @ Z3
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_369_prop__restrict,axiom,
! [X4: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X4 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_370_conj__subset__def,axiom,
! [A: set_set_nat,P: set_nat > $o,Q3: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P ) )
& ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_371_conj__subset__def,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o,Q3: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ P ) )
& ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_372_conj__subset__def,axiom,
! [A: set_na7556516505497143492la_a_b,P: ( nat > relational_fmla_a_b ) > $o,Q3: ( nat > relational_fmla_a_b ) > $o] :
( ( ord_le3065308906375977252la_a_b @ A
@ ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le3065308906375977252la_a_b @ A @ ( collec4193374254315349731la_a_b @ P ) )
& ( ord_le3065308906375977252la_a_b @ A @ ( collec4193374254315349731la_a_b @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_373_conj__subset__def,axiom,
! [A: set_Re5563258083572488516_b_nat,P: ( relational_fmla_a_b > nat ) > $o,Q3: ( relational_fmla_a_b > nat ) > $o] :
( ( ord_le1072050484451322276_b_nat @ A
@ ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le1072050484451322276_b_nat @ A @ ( collec115744212963325795_b_nat @ P ) )
& ( ord_le1072050484451322276_b_nat @ A @ ( collec115744212963325795_b_nat @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_374_conj__subset__def,axiom,
! [A: set_Re1288005135514575379la_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,Q3: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( ord_le116059350455505523la_a_b @ A
@ ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le116059350455505523la_a_b @ A @ ( collec5041345499257167282la_a_b @ P ) )
& ( ord_le116059350455505523la_a_b @ A @ ( collec5041345499257167282la_a_b @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_375_conj__subset__def,axiom,
! [A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ A
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
& ( ord_le4112832032246704949la_a_b @ A @ ( collec3419995626248312948la_a_b @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_376_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q3 @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_377_qp__gen,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_qp_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_gen_a_b @ X4 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% qp_gen
thf(fact_378_GreatestI2__order,axiom,
! [P: set_nat > $o,X4: set_nat,Q3: set_nat > $o] :
( ( P @ X4 )
=> ( ! [Y: set_nat] :
( ( P @ Y )
=> ( ord_less_eq_set_nat @ Y @ X4 ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( ord_less_eq_set_nat @ Y6 @ X3 ) )
=> ( Q3 @ X3 ) ) )
=> ( Q3 @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_379_GreatestI2__order,axiom,
! [P: set_Re381260168593705685la_a_b > $o,X4: set_Re381260168593705685la_a_b,Q3: set_Re381260168593705685la_a_b > $o] :
( ( P @ X4 )
=> ( ! [Y: set_Re381260168593705685la_a_b] :
( ( P @ Y )
=> ( ord_le4112832032246704949la_a_b @ Y @ X4 ) )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( P @ X3 )
=> ( ! [Y6: set_Re381260168593705685la_a_b] :
( ( P @ Y6 )
=> ( ord_le4112832032246704949la_a_b @ Y6 @ X3 ) )
=> ( Q3 @ X3 ) ) )
=> ( Q3 @ ( order_6069613389275354556la_a_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_380_GreatestI2__order,axiom,
! [P: ( nat > $o ) > $o,X4: nat > $o,Q3: ( nat > $o ) > $o] :
( ( P @ X4 )
=> ( ! [Y: nat > $o] :
( ( P @ Y )
=> ( ord_less_eq_nat_o @ Y @ X4 ) )
=> ( ! [X3: nat > $o] :
( ( P @ X3 )
=> ( ! [Y6: nat > $o] :
( ( P @ Y6 )
=> ( ord_less_eq_nat_o @ Y6 @ X3 ) )
=> ( Q3 @ X3 ) ) )
=> ( Q3 @ ( order_Greatest_nat_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_381_GreatestI2__order,axiom,
! [P: ( relational_fmla_a_b > $o ) > $o,X4: relational_fmla_a_b > $o,Q3: ( relational_fmla_a_b > $o ) > $o] :
( ( P @ X4 )
=> ( ! [Y: relational_fmla_a_b > $o] :
( ( P @ Y )
=> ( ord_le7191224889845164944_a_b_o @ Y @ X4 ) )
=> ( ! [X3: relational_fmla_a_b > $o] :
( ( P @ X3 )
=> ( ! [Y6: relational_fmla_a_b > $o] :
( ( P @ Y6 )
=> ( ord_le7191224889845164944_a_b_o @ Y6 @ X3 ) )
=> ( Q3 @ X3 ) ) )
=> ( Q3 @ ( order_7602139253572565961_a_b_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_382_GreatestI2__order,axiom,
! [P: nat > $o,X4: nat,Q3: nat > $o] :
( ( P @ X4 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) )
=> ( Q3 @ X3 ) ) )
=> ( Q3 @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_383_Greatest__equality,axiom,
! [P: set_nat > $o,X4: set_nat] :
( ( P @ X4 )
=> ( ! [Y: set_nat] :
( ( P @ Y )
=> ( ord_less_eq_set_nat @ Y @ X4 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_384_Greatest__equality,axiom,
! [P: set_Re381260168593705685la_a_b > $o,X4: set_Re381260168593705685la_a_b] :
( ( P @ X4 )
=> ( ! [Y: set_Re381260168593705685la_a_b] :
( ( P @ Y )
=> ( ord_le4112832032246704949la_a_b @ Y @ X4 ) )
=> ( ( order_6069613389275354556la_a_b @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_385_Greatest__equality,axiom,
! [P: ( nat > $o ) > $o,X4: nat > $o] :
( ( P @ X4 )
=> ( ! [Y: nat > $o] :
( ( P @ Y )
=> ( ord_less_eq_nat_o @ Y @ X4 ) )
=> ( ( order_Greatest_nat_o @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_386_Greatest__equality,axiom,
! [P: ( relational_fmla_a_b > $o ) > $o,X4: relational_fmla_a_b > $o] :
( ( P @ X4 )
=> ( ! [Y: relational_fmla_a_b > $o] :
( ( P @ Y )
=> ( ord_le7191224889845164944_a_b_o @ Y @ X4 ) )
=> ( ( order_7602139253572565961_a_b_o @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_387_Greatest__equality,axiom,
! [P: nat > $o,X4: nat] :
( ( P @ X4 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( order_Greatest_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_388_is__singletonI,axiom,
! [X4: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_389_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_390_insert__absorb2,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X4 @ ( insert7010464514620295119la_a_b @ X4 @ A ) )
= ( insert7010464514620295119la_a_b @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_391_insert__absorb2,axiom,
! [X4: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ X4 @ A ) )
= ( insert_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_392_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_393_insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_394_insert__iff,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_395_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_396_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_397_insertCI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_398_insertCI,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_399_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_400_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_401_singletonI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) ).
% singletonI
thf(fact_402_singletonI,axiom,
! [A2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_403_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_404_insert__subset,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( ( member_set_nat @ X4 @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_405_insert__subset,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ A ) @ B )
= ( ( member3481406638322139244la_a_b @ X4 @ B )
& ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_406_insert__subset,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ B )
= ( ( member4680049679412964150la_a_b @ X4 @ B )
& ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_407_insert__subset,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_408_singleton__conv2,axiom,
! [A2: set_nat] :
( ( collect_set_nat
@ ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_409_singleton__conv2,axiom,
! [A2: nat > nat] :
( ( collect_nat_nat
@ ( ^ [Y4: nat > nat,Z2: nat > nat] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% singleton_conv2
thf(fact_410_singleton__conv2,axiom,
! [A2: nat > relational_fmla_a_b] :
( ( collec4193374254315349731la_a_b
@ ( ^ [Y4: nat > relational_fmla_a_b,Z2: nat > relational_fmla_a_b] : ( Y4 = Z2 )
@ A2 ) )
= ( insert6993385188428902846la_a_b @ A2 @ bot_bo5171125281866047832la_a_b ) ) ).
% singleton_conv2
thf(fact_411_singleton__conv2,axiom,
! [A2: relational_fmla_a_b > nat] :
( ( collec115744212963325795_b_nat
@ ( ^ [Y4: relational_fmla_a_b > nat,Z2: relational_fmla_a_b > nat] : ( Y4 = Z2 )
@ A2 ) )
= ( insert2915755147076878910_b_nat @ A2 @ bot_bo3177866859941392856_b_nat ) ) ).
% singleton_conv2
thf(fact_412_singleton__conv2,axiom,
! [A2: relational_fmla_a_b > relational_fmla_a_b] :
( ( collec5041345499257167282la_a_b
@ ( ^ [Y4: relational_fmla_a_b > relational_fmla_a_b,Z2: relational_fmla_a_b > relational_fmla_a_b] : ( Y4 = Z2 )
@ A2 ) )
= ( insert8904949763332019597la_a_b @ A2 @ bot_bo9179849999556691623la_a_b ) ) ).
% singleton_conv2
thf(fact_413_singleton__conv2,axiom,
! [A2: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ( ^ [Y4: relational_fmla_a_b,Z2: relational_fmla_a_b] : ( Y4 = Z2 )
@ A2 ) )
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv2
thf(fact_414_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_415_singleton__conv,axiom,
! [A2: set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( X = A2 ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_416_singleton__conv,axiom,
! [A2: nat > nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( X = A2 ) )
= ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% singleton_conv
thf(fact_417_singleton__conv,axiom,
! [A2: nat > relational_fmla_a_b] :
( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] : ( X = A2 ) )
= ( insert6993385188428902846la_a_b @ A2 @ bot_bo5171125281866047832la_a_b ) ) ).
% singleton_conv
thf(fact_418_singleton__conv,axiom,
! [A2: relational_fmla_a_b > nat] :
( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] : ( X = A2 ) )
= ( insert2915755147076878910_b_nat @ A2 @ bot_bo3177866859941392856_b_nat ) ) ).
% singleton_conv
thf(fact_419_singleton__conv,axiom,
! [A2: relational_fmla_a_b > relational_fmla_a_b] :
( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( X = A2 ) )
= ( insert8904949763332019597la_a_b @ A2 @ bot_bo9179849999556691623la_a_b ) ) ).
% singleton_conv
thf(fact_420_singleton__conv,axiom,
! [A2: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( X = A2 ) )
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv
thf(fact_421_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_422_eq__or__mem__image__simp,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_nat @ L @ B ) ) ) )
= ( insert_nat @ ( F @ A2 )
@ ( collect_nat
@ ^ [Uu: nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( member_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_423_eq__or__mem__image__simp,axiom,
! [F: nat > relational_fmla_a_b,A2: nat,B: set_nat] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_nat @ L @ B ) ) ) )
= ( insert7010464514620295119la_a_b @ ( F @ A2 )
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( member_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_424_eq__or__mem__image__simp,axiom,
! [F: relational_fmla_a_b > nat,A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( collect_nat
@ ^ [Uu: nat] :
? [L: relational_fmla_a_b] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member4680049679412964150la_a_b @ L @ B ) ) ) )
= ( insert_nat @ ( F @ A2 )
@ ( collect_nat
@ ^ [Uu: nat] :
? [L: relational_fmla_a_b] :
( ( Uu
= ( F @ L ) )
& ( member4680049679412964150la_a_b @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_425_eq__or__mem__image__simp,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: relational_fmla_a_b] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member4680049679412964150la_a_b @ L @ B ) ) ) )
= ( insert7010464514620295119la_a_b @ ( F @ A2 )
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: relational_fmla_a_b] :
( ( Uu
= ( F @ L ) )
& ( member4680049679412964150la_a_b @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_426_eq__or__mem__image__simp,axiom,
! [F: set_nat > nat,A2: set_nat,B: set_set_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_set_nat @ L @ B ) ) ) )
= ( insert_nat @ ( F @ A2 )
@ ( collect_nat
@ ^ [Uu: nat] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( member_set_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_427_eq__or__mem__image__simp,axiom,
! [F: nat > set_nat,A2: nat,B: set_nat] :
( ( collect_set_nat
@ ^ [Uu: set_nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_nat @ L @ B ) ) ) )
= ( insert_set_nat @ ( F @ A2 )
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( member_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_428_eq__or__mem__image__simp,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( collect_set_nat
@ ^ [Uu: set_nat] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_set_nat @ L @ B ) ) ) )
= ( insert_set_nat @ ( F @ A2 )
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( member_set_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_429_eq__or__mem__image__simp,axiom,
! [F: nat > nat > nat,A2: nat,B: set_nat] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_nat @ L @ B ) ) ) )
= ( insert_nat_nat @ ( F @ A2 )
@ ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [L: nat] :
( ( Uu
= ( F @ L ) )
& ( member_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_430_eq__or__mem__image__simp,axiom,
! [F: set_nat > relational_fmla_a_b,A2: set_nat,B: set_set_nat] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member_set_nat @ L @ B ) ) ) )
= ( insert7010464514620295119la_a_b @ ( F @ A2 )
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [L: set_nat] :
( ( Uu
= ( F @ L ) )
& ( member_set_nat @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_431_eq__or__mem__image__simp,axiom,
! [F: set_Re381260168593705685la_a_b > nat,A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( collect_nat
@ ^ [Uu: nat] :
? [L: set_Re381260168593705685la_a_b] :
( ( Uu
= ( F @ L ) )
& ( ( L = A2 )
| ( member3481406638322139244la_a_b @ L @ B ) ) ) )
= ( insert_nat @ ( F @ A2 )
@ ( collect_nat
@ ^ [Uu: nat] :
? [L: set_Re381260168593705685la_a_b] :
( ( Uu
= ( F @ L ) )
& ( member3481406638322139244la_a_b @ L @ B ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_432_singleton__insert__inj__eq,axiom,
! [B2: relational_fmla_a_b,A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_433_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_434_singleton__insert__inj__eq_H,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ A )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( A2 = B2 )
& ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_435_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_436_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B6: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B6 ) )
& ~ ( member_set_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_437_mk__disjoint__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ? [B6: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ A2 @ B6 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_438_mk__disjoint__insert,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [B6: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ A2 @ B6 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_439_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat @ A2 @ B6 ) )
& ~ ( member_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_440_insert__commute,axiom,
! [X4: relational_fmla_a_b,Y3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X4 @ ( insert7010464514620295119la_a_b @ Y3 @ A ) )
= ( insert7010464514620295119la_a_b @ Y3 @ ( insert7010464514620295119la_a_b @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_441_insert__commute,axiom,
! [X4: nat,Y3: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ Y3 @ A ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_442_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B2 @ B )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B2 @ C3 ) )
& ~ ( member_set_nat @ B2 @ C3 )
& ( B
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_443_insert__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ~ ( member3481406638322139244la_a_b @ B2 @ B )
=> ( ( ( insert2023870700798818565la_a_b @ A2 @ A )
= ( insert2023870700798818565la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ B2 @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ B2 @ C3 )
& ( B
= ( insert2023870700798818565la_a_b @ A2 @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_444_insert__eq__iff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ A2 @ A )
= ( insert7010464514620295119la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ B2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ B2 @ C3 )
& ( B
= ( insert7010464514620295119la_a_b @ A2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_445_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_446_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_447_insert__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( insert2023870700798818565la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_448_insert__absorb,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_449_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_450_insert__ident,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ~ ( member_set_nat @ X4 @ B )
=> ( ( ( insert_set_nat @ X4 @ A )
= ( insert_set_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_451_insert__ident,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ~ ( member3481406638322139244la_a_b @ X4 @ B )
=> ( ( ( insert2023870700798818565la_a_b @ X4 @ A )
= ( insert2023870700798818565la_a_b @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_452_insert__ident,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X4 @ A )
= ( insert7010464514620295119la_a_b @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_453_insert__ident,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ~ ( member_nat @ X4 @ B )
=> ( ( ( insert_nat @ X4 @ A )
= ( insert_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_454_Set_Oset__insert,axiom,
! [X4: set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ! [B6: set_set_nat] :
( ( A
= ( insert_set_nat @ X4 @ B6 ) )
=> ( member_set_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_455_Set_Oset__insert,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ A )
=> ~ ! [B6: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ X4 @ B6 ) )
=> ( member3481406638322139244la_a_b @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_456_Set_Oset__insert,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ~ ! [B6: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ X4 @ B6 ) )
=> ( member4680049679412964150la_a_b @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_457_Set_Oset__insert,axiom,
! [X4: nat,A: set_nat] :
( ( member_nat @ X4 @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat @ X4 @ B6 ) )
=> ( member_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_458_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ B )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_459_insertI2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ B )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_460_insertI2,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ B )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_461_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_462_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_463_insertI1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_464_insertI1,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_465_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_466_insertE,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_467_insertE,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_468_insertE,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_469_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_470_insert__subsetI,axiom,
! [X4: set_nat,A: set_set_nat,X5: set_set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ A )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_471_insert__subsetI,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,X5: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ( ord_le1577343677690852715la_a_b @ X5 @ A )
=> ( ord_le1577343677690852715la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_472_insert__subsetI,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ X5 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_473_insert__subsetI,axiom,
! [X4: nat,A: set_nat,X5: set_nat] :
( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_474_insert__compr,axiom,
( insert2023870700798818565la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( X = A5 )
| ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_475_insert__compr,axiom,
( insert_set_nat
= ( ^ [A5: set_nat,B5: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A5 )
| ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_476_insert__compr,axiom,
( insert_nat_nat
= ( ^ [A5: nat > nat,B5: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A5 )
| ( member_nat_nat @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_477_insert__compr,axiom,
( insert6993385188428902846la_a_b
= ( ^ [A5: nat > relational_fmla_a_b,B5: set_na7556516505497143492la_a_b] :
( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( X = A5 )
| ( member8923333377441230501la_a_b @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_478_insert__compr,axiom,
( insert2915755147076878910_b_nat
= ( ^ [A5: relational_fmla_a_b > nat,B5: set_Re5563258083572488516_b_nat] :
( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( X = A5 )
| ( member4845703336089206565_b_nat @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_479_insert__compr,axiom,
( insert8904949763332019597la_a_b
= ( ^ [A5: relational_fmla_a_b > relational_fmla_a_b,B5: set_Re1288005135514575379la_a_b] :
( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( X = A5 )
| ( member8433577210552456052la_a_b @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_480_insert__compr,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A5: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A5 )
| ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_481_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A5 )
| ( member_nat @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_482_insert__Collect,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( insert_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( collect_set_nat
@ ^ [U: set_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_483_insert__Collect,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( insert_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( collect_nat_nat
@ ^ [U: nat > nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_484_insert__Collect,axiom,
! [A2: nat > relational_fmla_a_b,P: ( nat > relational_fmla_a_b ) > $o] :
( ( insert6993385188428902846la_a_b @ A2 @ ( collec4193374254315349731la_a_b @ P ) )
= ( collec4193374254315349731la_a_b
@ ^ [U: nat > relational_fmla_a_b] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_485_insert__Collect,axiom,
! [A2: relational_fmla_a_b > nat,P: ( relational_fmla_a_b > nat ) > $o] :
( ( insert2915755147076878910_b_nat @ A2 @ ( collec115744212963325795_b_nat @ P ) )
= ( collec115744212963325795_b_nat
@ ^ [U: relational_fmla_a_b > nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_486_insert__Collect,axiom,
! [A2: relational_fmla_a_b > relational_fmla_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( insert8904949763332019597la_a_b @ A2 @ ( collec5041345499257167282la_a_b @ P ) )
= ( collec5041345499257167282la_a_b
@ ^ [U: relational_fmla_a_b > relational_fmla_a_b] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_487_insert__Collect,axiom,
! [A2: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( insert7010464514620295119la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) )
= ( collec3419995626248312948la_a_b
@ ^ [U: relational_fmla_a_b] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_488_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_489_singleton__inject,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_490_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_491_insert__not__empty,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ A )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_492_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_493_doubleton__eq__iff,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,C: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_494_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_495_singleton__iff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_496_singleton__iff,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_497_singleton__iff,axiom,
! [B2: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_498_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_499_singletonD,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_500_singletonD,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_501_singletonD,axiom,
! [B2: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_502_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_503_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_504_subset__insertI2,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_505_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_506_subset__insertI,axiom,
! [B: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ B ) ) ).
% subset_insertI
thf(fact_507_subset__insert,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X4 @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_508_subset__insert,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ( ord_le1577343677690852715la_a_b @ A @ ( insert2023870700798818565la_a_b @ X4 @ B ) )
= ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_509_subset__insert,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ B ) )
= ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_510_subset__insert,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_511_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_512_insert__mono,axiom,
! [C2: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ C2 ) @ ( insert7010464514620295119la_a_b @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_513_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ R )
@ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_514_pred__subset__eq,axiom,
! [R: set_se6865892389300016395la_a_b,S: set_se6865892389300016395la_a_b] :
( ( ord_le8511361905821865178_a_b_o
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ R )
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ S ) )
= ( ord_le1577343677690852715la_a_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_515_pred__subset__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( ord_le7191224889845164944_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ R )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ S ) )
= ( ord_le4112832032246704949la_a_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_516_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_517_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_518_less__eq__set__def,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( ord_le8511361905821865178_a_b_o
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A3 )
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_519_less__eq__set__def,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( ord_le7191224889845164944_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A3 )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_520_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_521_Collect__conv__if2,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_522_Collect__conv__if2,axiom,
! [P: ( nat > nat ) > $o,A2: nat > nat] :
( ( ( P @ A2 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_nat_nat ) ) ) ).
% Collect_conv_if2
thf(fact_523_Collect__conv__if2,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o,A2: nat > relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert6993385188428902846la_a_b @ A2 @ bot_bo5171125281866047832la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bo5171125281866047832la_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_524_Collect__conv__if2,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o,A2: relational_fmla_a_b > nat] :
( ( ( P @ A2 )
=> ( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert2915755147076878910_b_nat @ A2 @ bot_bo3177866859941392856_b_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bo3177866859941392856_b_nat ) ) ) ).
% Collect_conv_if2
thf(fact_525_Collect__conv__if2,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,A2: relational_fmla_a_b > relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert8904949763332019597la_a_b @ A2 @ bot_bo9179849999556691623la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bo9179849999556691623la_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_526_Collect__conv__if2,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_527_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_528_Collect__conv__if,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_529_Collect__conv__if,axiom,
! [P: ( nat > nat ) > $o,A2: nat > nat] :
( ( ( P @ A2 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_nat_nat ) ) ) ).
% Collect_conv_if
thf(fact_530_Collect__conv__if,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o,A2: nat > relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert6993385188428902846la_a_b @ A2 @ bot_bo5171125281866047832la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bo5171125281866047832la_a_b ) ) ) ).
% Collect_conv_if
thf(fact_531_Collect__conv__if,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o,A2: relational_fmla_a_b > nat] :
( ( ( P @ A2 )
=> ( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert2915755147076878910_b_nat @ A2 @ bot_bo3177866859941392856_b_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bo3177866859941392856_b_nat ) ) ) ).
% Collect_conv_if
thf(fact_532_Collect__conv__if,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,A2: relational_fmla_a_b > relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert8904949763332019597la_a_b @ A2 @ bot_bo9179849999556691623la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bo9179849999556691623la_a_b ) ) ) ).
% Collect_conv_if
thf(fact_533_Collect__conv__if,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ( ( P @ A2 )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if
thf(fact_534_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_535_subset__singletonD,axiom,
! [A: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( A = bot_bo4495933725496725865la_a_b )
| ( A
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_536_subset__singletonD,axiom,
! [A: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_537_subset__singleton__iff,axiom,
! [X5: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( X5 = bot_bo4495933725496725865la_a_b )
| ( X5
= ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_538_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_539_is__singletonE,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( is_sin6594375743535830443la_a_b @ A )
=> ~ ! [X3: relational_fmla_a_b] :
( A
!= ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% is_singletonE
thf(fact_540_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_541_is__singleton__def,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
? [X: relational_fmla_a_b] :
( A3
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_542_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_543_order__mono__setup_Omono__if,axiom,
! [T2: set_nat,T3: set_nat,E: set_nat,E2: set_nat,B2: $o] :
( ( ord_less_eq_set_nat @ T2 @ T3 )
=> ( ( ord_less_eq_set_nat @ E @ E2 )
=> ( ord_less_eq_set_nat @ ( if_set_nat @ B2 @ T2 @ E ) @ ( if_set_nat @ B2 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_544_order__mono__setup_Omono__if,axiom,
! [T2: set_Re381260168593705685la_a_b,T3: set_Re381260168593705685la_a_b,E: set_Re381260168593705685la_a_b,E2: set_Re381260168593705685la_a_b,B2: $o] :
( ( ord_le4112832032246704949la_a_b @ T2 @ T3 )
=> ( ( ord_le4112832032246704949la_a_b @ E @ E2 )
=> ( ord_le4112832032246704949la_a_b @ ( if_set2835548578466827919la_a_b @ B2 @ T2 @ E ) @ ( if_set2835548578466827919la_a_b @ B2 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_545_order__mono__setup_Omono__if,axiom,
! [T2: nat > $o,T3: nat > $o,E: nat > $o,E2: nat > $o,B2: $o] :
( ( ord_less_eq_nat_o @ T2 @ T3 )
=> ( ( ord_less_eq_nat_o @ E @ E2 )
=> ( ord_less_eq_nat_o @ ( if_nat_o @ B2 @ T2 @ E ) @ ( if_nat_o @ B2 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_546_order__mono__setup_Omono__if,axiom,
! [T2: relational_fmla_a_b > $o,T3: relational_fmla_a_b > $o,E: relational_fmla_a_b > $o,E2: relational_fmla_a_b > $o,B2: $o] :
( ( ord_le7191224889845164944_a_b_o @ T2 @ T3 )
=> ( ( ord_le7191224889845164944_a_b_o @ E @ E2 )
=> ( ord_le7191224889845164944_a_b_o @ ( if_Rel8262680441166333110_a_b_o @ B2 @ T2 @ E ) @ ( if_Rel8262680441166333110_a_b_o @ B2 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_547_order__mono__setup_Omono__if,axiom,
! [T2: nat,T3: nat,E: nat,E2: nat,B2: $o] :
( ( ord_less_eq_nat @ T2 @ T3 )
=> ( ( ord_less_eq_nat @ E @ E2 )
=> ( ord_less_eq_nat @ ( if_nat @ B2 @ T2 @ E ) @ ( if_nat @ B2 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_548_is__singleton__the__elem,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( A3
= ( insert7010464514620295119la_a_b @ ( the_el6350558617753882986la_a_b @ A3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_549_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_550_the__elem__eq,axiom,
! [X4: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
= X4 ) ).
% the_elem_eq
thf(fact_551_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_552_subset__singleton__iff__Uniq,axiom,
! [A: set_set_nat] :
( ( ? [A5: set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ A5 @ bot_bot_set_set_nat ) ) )
= ( uniq_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_553_subset__singleton__iff__Uniq,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ? [A5: set_Re381260168593705685la_a_b] : ( ord_le1577343677690852715la_a_b @ A @ ( insert2023870700798818565la_a_b @ A5 @ bot_bo2891247006866115487la_a_b ) ) )
= ( uniq_s7602024809403740688la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_554_subset__singleton__iff__Uniq,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ? [A5: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ A5 @ bot_bo4495933725496725865la_a_b ) ) )
= ( uniq_R5111430000950013146la_a_b
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_555_subset__singleton__iff__Uniq,axiom,
! [A: set_nat] :
( ( ? [A5: nat] : ( ord_less_eq_set_nat @ A @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_556_cov_H_Oap,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_ap_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_cov_a_b2 @ X4 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov'.ap
thf(fact_557_gen_Ointros_I2_J,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_ap_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_gen_a_b @ X4 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen.intros(2)
thf(fact_558_flat__Disj_Osimps_I3_J,axiom,
! [V: $o] :
( ( restri569617705344514291sj_a_b @ ( relational_Bool_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(3)
thf(fact_559_subset__Compl__singleton,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_560_subset__Compl__singleton,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ ( uminus5378734866260624980la_a_b @ ( insert2023870700798818565la_a_b @ B2 @ bot_bo2891247006866115487la_a_b ) ) )
= ( ~ ( member3481406638322139244la_a_b @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_561_subset__Compl__singleton,axiom,
! [A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( uminus235785408959094814la_a_b @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_562_subset__Compl__singleton,axiom,
! [A: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_563_sub_Osimps_I1_J,axiom,
! [T2: $o] :
( ( relational_sub_a_b @ ( relational_Bool_a_b @ T2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(1)
thf(fact_564_relprop__triggers_I6_J,axiom,
! [R: set_nat,R2: set_nat] :
( ( ord_less_eq_set_nat @ R @ R2 )
=> ( ord_less_eq_set_nat @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_565_relprop__triggers_I6_J,axiom,
! [R: set_Re381260168593705685la_a_b,R2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ R @ R2 )
=> ( ord_le4112832032246704949la_a_b @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_566_ComplI,axiom,
! [C: set_nat,A: set_set_nat] :
( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) ) ) ).
% ComplI
thf(fact_567_ComplI,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ ( uminus5378734866260624980la_a_b @ A ) ) ) ).
% ComplI
thf(fact_568_ComplI,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ ( uminus235785408959094814la_a_b @ A ) ) ) ).
% ComplI
thf(fact_569_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_570_Compl__iff,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
= ( ~ ( member_set_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_571_Compl__iff,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( uminus5378734866260624980la_a_b @ A ) )
= ( ~ ( member3481406638322139244la_a_b @ C @ A ) ) ) ).
% Compl_iff
thf(fact_572_Compl__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( uminus235785408959094814la_a_b @ A ) )
= ( ~ ( member4680049679412964150la_a_b @ C @ A ) ) ) ).
% Compl_iff
thf(fact_573_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_574_Compl__eq__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( uminus5710092332889474511et_nat @ A )
= ( uminus5710092332889474511et_nat @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_575_Compl__eq__Compl__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( uminus235785408959094814la_a_b @ A )
= ( uminus235785408959094814la_a_b @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_576_predicate1I,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_less_eq_nat_o @ P @ Q3 ) ) ).
% predicate1I
thf(fact_577_predicate1I,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le7191224889845164944_a_b_o @ P @ Q3 ) ) ).
% predicate1I
thf(fact_578_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_579_Compl__anti__mono,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ B ) @ ( uminus235785408959094814la_a_b @ A ) ) ) ).
% Compl_anti_mono
thf(fact_580_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_581_Compl__subset__Compl__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ A ) @ ( uminus235785408959094814la_a_b @ B ) )
= ( ord_le4112832032246704949la_a_b @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_582_predicate1D,axiom,
! [P: nat > $o,Q3: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ P @ Q3 )
=> ( ( P @ X4 )
=> ( Q3 @ X4 ) ) ) ).
% predicate1D
thf(fact_583_predicate1D,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o,X4: relational_fmla_a_b] :
( ( ord_le7191224889845164944_a_b_o @ P @ Q3 )
=> ( ( P @ X4 )
=> ( Q3 @ X4 ) ) ) ).
% predicate1D
thf(fact_584_rev__predicate1D,axiom,
! [P: nat > $o,X4: nat,Q3: nat > $o] :
( ( P @ X4 )
=> ( ( ord_less_eq_nat_o @ P @ Q3 )
=> ( Q3 @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_585_rev__predicate1D,axiom,
! [P: relational_fmla_a_b > $o,X4: relational_fmla_a_b,Q3: relational_fmla_a_b > $o] :
( ( P @ X4 )
=> ( ( ord_le7191224889845164944_a_b_o @ P @ Q3 )
=> ( Q3 @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_586_flat__Disj__sub,axiom,
! [Q3: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ ( restri569617705344514291sj_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ).
% flat_Disj_sub
thf(fact_587_ComplD,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
=> ~ ( member_set_nat @ C @ A ) ) ).
% ComplD
thf(fact_588_ComplD,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( uminus5378734866260624980la_a_b @ A ) )
=> ~ ( member3481406638322139244la_a_b @ C @ A ) ) ).
% ComplD
thf(fact_589_ComplD,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( uminus235785408959094814la_a_b @ A ) )
=> ~ ( member4680049679412964150la_a_b @ C @ A ) ) ).
% ComplD
thf(fact_590_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_591_double__complement,axiom,
! [A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ A ) )
= A ) ).
% double_complement
thf(fact_592_double__complement,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( uminus235785408959094814la_a_b @ A ) )
= A ) ).
% double_complement
thf(fact_593_Collect__neg__eq,axiom,
! [P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
~ ( P @ X ) )
= ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_594_Collect__neg__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X: nat > nat] :
~ ( P @ X ) )
= ( uminus4145589374814813630at_nat @ ( collect_nat_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_595_Collect__neg__eq,axiom,
! [P: ( nat > relational_fmla_a_b ) > $o] :
( ( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
~ ( P @ X ) )
= ( uminus4144662930026986637la_a_b @ ( collec4193374254315349731la_a_b @ P ) ) ) ).
% Collect_neg_eq
thf(fact_596_Collect__neg__eq,axiom,
! [P: ( relational_fmla_a_b > nat ) > $o] :
( ( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
~ ( P @ X ) )
= ( uminus2151404508102331661_b_nat @ ( collec115744212963325795_b_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_597_Collect__neg__eq,axiom,
! [P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
( ( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
~ ( P @ X ) )
= ( uminus8744632006355346652la_a_b @ ( collec5041345499257167282la_a_b @ P ) ) ) ).
% Collect_neg_eq
thf(fact_598_Collect__neg__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
~ ( P @ X ) )
= ( uminus235785408959094814la_a_b @ ( collec3419995626248312948la_a_b @ P ) ) ) ).
% Collect_neg_eq
thf(fact_599_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_600_Compl__eq,axiom,
( uminus5378734866260624980la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_601_Compl__eq,axiom,
( uminus613421341184616069et_nat
= ( ^ [A3: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
~ ( member_set_nat @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_602_Compl__eq,axiom,
( uminus4145589374814813630at_nat
= ( ^ [A3: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
~ ( member_nat_nat @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_603_Compl__eq,axiom,
( uminus4144662930026986637la_a_b
= ( ^ [A3: set_na7556516505497143492la_a_b] :
( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
~ ( member8923333377441230501la_a_b @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_604_Compl__eq,axiom,
( uminus2151404508102331661_b_nat
= ( ^ [A3: set_Re5563258083572488516_b_nat] :
( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
~ ( member4845703336089206565_b_nat @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_605_Compl__eq,axiom,
( uminus8744632006355346652la_a_b
= ( ^ [A3: set_Re1288005135514575379la_a_b] :
( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
~ ( member8433577210552456052la_a_b @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_606_Compl__eq,axiom,
( uminus235785408959094814la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_607_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ^ [X: nat] :
~ ( member_nat @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_608_qp_Oap,axiom,
! [Q3: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q3 )
=> ( relational_qp_a_b @ Q3 ) ) ).
% qp.ap
thf(fact_609_fv__flat__DisjD,axiom,
! [Q5: relational_fmla_a_b,Q3: relational_fmla_a_b,X4: nat] :
( ( member4680049679412964150la_a_b @ Q5 @ ( restri569617705344514291sj_a_b @ Q3 ) )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q5 ) )
=> ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) ) ) ) ).
% fv_flat_DisjD
thf(fact_610_subset__Compl__self__eq,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( uminus235785408959094814la_a_b @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% subset_Compl_self_eq
thf(fact_611_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_612_compl__le__compl__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( uminus5710092332889474511et_nat @ Y3 ) )
= ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_613_compl__le__compl__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ X4 ) @ ( uminus235785408959094814la_a_b @ Y3 ) )
= ( ord_le4112832032246704949la_a_b @ Y3 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_614_compl__le__compl__iff,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ X4 ) @ ( uminus_uminus_nat_o @ Y3 ) )
= ( ord_less_eq_nat_o @ Y3 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_615_compl__le__compl__iff,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ ( uminus4297745234618001383_a_b_o @ X4 ) @ ( uminus4297745234618001383_a_b_o @ Y3 ) )
= ( ord_le7191224889845164944_a_b_o @ Y3 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_616_uminus__apply,axiom,
( uminus_uminus_nat_o
= ( ^ [A3: nat > $o,X: nat] : ( uminus_uminus_o @ ( A3 @ X ) ) ) ) ).
% uminus_apply
thf(fact_617_uminus__apply,axiom,
( uminus4297745234618001383_a_b_o
= ( ^ [A3: relational_fmla_a_b > $o,X: relational_fmla_a_b] : ( uminus_uminus_o @ ( A3 @ X ) ) ) ) ).
% uminus_apply
thf(fact_618_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( uminus5710092332889474511et_nat @ X4 )
= ( uminus5710092332889474511et_nat @ Y3 ) )
= ( X4 = Y3 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_619_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( uminus235785408959094814la_a_b @ X4 )
= ( uminus235785408959094814la_a_b @ Y3 ) )
= ( X4 = Y3 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_620_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( ( uminus_uminus_nat_o @ X4 )
= ( uminus_uminus_nat_o @ Y3 ) )
= ( X4 = Y3 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_621_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( ( uminus4297745234618001383_a_b_o @ X4 )
= ( uminus4297745234618001383_a_b_o @ Y3 ) )
= ( X4 = Y3 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_622_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X4: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ X4 ) )
= X4 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_623_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X4: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( uminus235785408959094814la_a_b @ X4 ) )
= X4 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_624_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X4: nat > $o] :
( ( uminus_uminus_nat_o @ ( uminus_uminus_nat_o @ X4 ) )
= X4 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_625_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X4: relational_fmla_a_b > $o] :
( ( uminus4297745234618001383_a_b_o @ ( uminus4297745234618001383_a_b_o @ X4 ) )
= X4 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_626_compl__le__swap2,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ X4 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_627_compl__le__swap2,axiom,
! [Y3: set_Re381260168593705685la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ Y3 ) @ X4 )
=> ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ X4 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_628_compl__le__swap2,axiom,
! [Y3: nat > $o,X4: nat > $o] :
( ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ Y3 ) @ X4 )
=> ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ X4 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_629_compl__le__swap2,axiom,
! [Y3: relational_fmla_a_b > $o,X4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ ( uminus4297745234618001383_a_b_o @ Y3 ) @ X4 )
=> ( ord_le7191224889845164944_a_b_o @ ( uminus4297745234618001383_a_b_o @ X4 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_630_uminus__set__def,axiom,
( uminus5378734866260624980la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ( uminus333668244874753329_a_b_o
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_631_uminus__set__def,axiom,
( uminus613421341184616069et_nat
= ( ^ [A3: set_set_nat] :
( collect_set_nat
@ ( uminus6401447641752708672_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_632_uminus__set__def,axiom,
( uminus4145589374814813630at_nat
= ( ^ [A3: set_nat_nat] :
( collect_nat_nat
@ ( uminus500452819864336455_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_633_uminus__set__def,axiom,
( uminus4144662930026986637la_a_b
= ( ^ [A3: set_na7556516505497143492la_a_b] :
( collec4193374254315349731la_a_b
@ ( uminus1532552339427715128_a_b_o
@ ^ [X: nat > relational_fmla_a_b] : ( member8923333377441230501la_a_b @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_634_uminus__set__def,axiom,
( uminus2151404508102331661_b_nat
= ( ^ [A3: set_Re5563258083572488516_b_nat] :
( collec115744212963325795_b_nat
@ ( uminus8556098566306461624_nat_o
@ ^ [X: relational_fmla_a_b > nat] : ( member4845703336089206565_b_nat @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_635_uminus__set__def,axiom,
( uminus8744632006355346652la_a_b
= ( ^ [A3: set_Re1288005135514575379la_a_b] :
( collec5041345499257167282la_a_b
@ ( uminus6795872291137556137_a_b_o
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_636_uminus__set__def,axiom,
( uminus235785408959094814la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( uminus4297745234618001383_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_637_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_638_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_639_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_640_verit__comp__simplify1_I2_J,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_641_verit__comp__simplify1_I2_J,axiom,
! [A2: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_642_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_643_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_644_fun__Compl__def,axiom,
( uminus_uminus_nat_o
= ( ^ [A3: nat > $o,X: nat] : ( uminus_uminus_o @ ( A3 @ X ) ) ) ) ).
% fun_Compl_def
thf(fact_645_fun__Compl__def,axiom,
( uminus4297745234618001383_a_b_o
= ( ^ [A3: relational_fmla_a_b > $o,X: relational_fmla_a_b] : ( uminus_uminus_o @ ( A3 @ X ) ) ) ) ).
% fun_Compl_def
thf(fact_646_compl__mono,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ ( uminus5710092332889474511et_nat @ X4 ) ) ) ).
% compl_mono
thf(fact_647_compl__mono,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X4 @ Y3 )
=> ( ord_le4112832032246704949la_a_b @ ( uminus235785408959094814la_a_b @ Y3 ) @ ( uminus235785408959094814la_a_b @ X4 ) ) ) ).
% compl_mono
thf(fact_648_compl__mono,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y3 )
=> ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ Y3 ) @ ( uminus_uminus_nat_o @ X4 ) ) ) ).
% compl_mono
thf(fact_649_compl__mono,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 )
=> ( ord_le7191224889845164944_a_b_o @ ( uminus4297745234618001383_a_b_o @ Y3 ) @ ( uminus4297745234618001383_a_b_o @ X4 ) ) ) ).
% compl_mono
thf(fact_650_compl__le__swap1,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X4 ) )
=> ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_651_compl__le__swap1,axiom,
! [Y3: set_Re381260168593705685la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y3 @ ( uminus235785408959094814la_a_b @ X4 ) )
=> ( ord_le4112832032246704949la_a_b @ X4 @ ( uminus235785408959094814la_a_b @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_652_compl__le__swap1,axiom,
! [Y3: nat > $o,X4: nat > $o] :
( ( ord_less_eq_nat_o @ Y3 @ ( uminus_uminus_nat_o @ X4 ) )
=> ( ord_less_eq_nat_o @ X4 @ ( uminus_uminus_nat_o @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_653_compl__le__swap1,axiom,
! [Y3: relational_fmla_a_b > $o,X4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ Y3 @ ( uminus4297745234618001383_a_b_o @ X4 ) )
=> ( ord_le7191224889845164944_a_b_o @ X4 @ ( uminus4297745234618001383_a_b_o @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_654_alt__ex1E_H,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y: nat] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ~ ( ? [X_1: nat] : ( P @ X_1 )
=> ~ ( uniq_nat @ P ) ) ) ).
% alt_ex1E'
thf(fact_655_alt__ex1E_H,axiom,
! [P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( P @ X6 )
& ! [Y: relational_fmla_a_b] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ~ ( ? [X_1: relational_fmla_a_b] : ( P @ X_1 )
=> ~ ( uniq_R5111430000950013146la_a_b @ P ) ) ) ).
% alt_ex1E'
thf(fact_656_ex1__iff__ex__Uniq,axiom,
( ex1_nat
= ( ^ [P2: nat > $o] :
( ? [X2: nat] : ( P2 @ X2 )
& ( uniq_nat @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_657_ex1__iff__ex__Uniq,axiom,
( ex1_Re433797028859071237la_a_b
= ( ^ [P2: relational_fmla_a_b > $o] :
( ? [X2: relational_fmla_a_b] : ( P2 @ X2 )
& ( uniq_R5111430000950013146la_a_b @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_658_gen_H_Ointros_I2_J,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_ap_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_gen_a_b2 @ X4 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen'.intros(2)
thf(fact_659_Uniq__D,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( uniq_nat @ P )
=> ( ( P @ A2 )
=> ( ( P @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% Uniq_D
thf(fact_660_Uniq__D,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( uniq_R5111430000950013146la_a_b @ P )
=> ( ( P @ A2 )
=> ( ( P @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% Uniq_D
thf(fact_661_Uniq__I,axiom,
! [P: nat > $o] :
( ! [X3: nat,Y: nat] :
( ( P @ X3 )
=> ( ( P @ Y )
=> ( Y = X3 ) ) )
=> ( uniq_nat @ P ) ) ).
% Uniq_I
thf(fact_662_Uniq__I,axiom,
! [P: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( P @ X3 )
=> ( ( P @ Y )
=> ( Y = X3 ) ) )
=> ( uniq_R5111430000950013146la_a_b @ P ) ) ).
% Uniq_I
thf(fact_663_Uniq__def,axiom,
( uniq_nat
= ( ^ [P2: nat > $o] :
! [X: nat,Y5: nat] :
( ( P2 @ X )
=> ( ( P2 @ Y5 )
=> ( Y5 = X ) ) ) ) ) ).
% Uniq_def
thf(fact_664_Uniq__def,axiom,
( uniq_R5111430000950013146la_a_b
= ( ^ [P2: relational_fmla_a_b > $o] :
! [X: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( P2 @ X )
=> ( ( P2 @ Y5 )
=> ( Y5 = X ) ) ) ) ) ).
% Uniq_def
thf(fact_665_the__elem__def,axiom,
( the_el6350558617753882986la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b] :
( the_Re7701088767514941656la_a_b
@ ^ [X: relational_fmla_a_b] :
( X2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% the_elem_def
thf(fact_666_the__elem__def,axiom,
( the_elem_nat
= ( ^ [X2: set_nat] :
( the_nat
@ ^ [X: nat] :
( X2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% the_elem_def
thf(fact_667_cov_Oap,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_ap_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_cov_a_b @ X4 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov.ap
thf(fact_668_the__sym__eq__trivial,axiom,
! [X4: nat] :
( ( the_nat
@ ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_669_the__sym__eq__trivial,axiom,
! [X4: relational_fmla_a_b] :
( ( the_Re7701088767514941656la_a_b
@ ( ^ [Y4: relational_fmla_a_b,Z2: relational_fmla_a_b] : ( Y4 = Z2 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_670_the__eq__trivial,axiom,
! [A2: nat] :
( ( the_nat
@ ^ [X: nat] : ( X = A2 ) )
= A2 ) ).
% the_eq_trivial
thf(fact_671_the__eq__trivial,axiom,
! [A2: relational_fmla_a_b] :
( ( the_Re7701088767514941656la_a_b
@ ^ [X: relational_fmla_a_b] : ( X = A2 ) )
= A2 ) ).
% the_eq_trivial
thf(fact_672_the__equality,axiom,
! [P: nat > $o,A2: nat] :
( ( P @ A2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the_nat @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_673_the__equality,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ( P @ A2 )
=> ( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the_Re7701088767514941656la_a_b @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_674_the1__equality,axiom,
! [P: nat > $o,A2: nat] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y: nat] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( ( P @ A2 )
=> ( ( the_nat @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_675_the1__equality,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ? [X6: relational_fmla_a_b] :
( ( P @ X6 )
& ! [Y: relational_fmla_a_b] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( ( P @ A2 )
=> ( ( the_Re7701088767514941656la_a_b @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_676_the1I2,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y: nat] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( Q3 @ ( the_nat @ P ) ) ) ) ).
% the1I2
thf(fact_677_the1I2,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( P @ X6 )
& ! [Y: relational_fmla_a_b] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( Q3 @ ( the_Re7701088767514941656la_a_b @ P ) ) ) ) ).
% the1I2
thf(fact_678_If__def,axiom,
( if_nat
= ( ^ [P2: $o,X: nat,Y5: nat] :
( the_nat
@ ^ [Z4: nat] :
( ( P2
=> ( Z4 = X ) )
& ( ~ P2
=> ( Z4 = Y5 ) ) ) ) ) ) ).
% If_def
thf(fact_679_If__def,axiom,
( if_Rel1279876242545935705la_a_b
= ( ^ [P2: $o,X: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( the_Re7701088767514941656la_a_b
@ ^ [Z4: relational_fmla_a_b] :
( ( P2
=> ( Z4 = X ) )
& ( ~ P2
=> ( Z4 = Y5 ) ) ) ) ) ) ).
% If_def
thf(fact_680_theI2,axiom,
! [P: nat > $o,A2: nat,Q3: nat > $o] :
( ( P @ A2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( Q3 @ ( the_nat @ P ) ) ) ) ) ).
% theI2
thf(fact_681_theI2,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b,Q3: relational_fmla_a_b > $o] :
( ( P @ A2 )
=> ( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( Q3 @ ( the_Re7701088767514941656la_a_b @ P ) ) ) ) ) ).
% theI2
thf(fact_682_theI_H,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y: nat] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( P @ ( the_nat @ P ) ) ) ).
% theI'
thf(fact_683_theI_H,axiom,
! [P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( P @ X6 )
& ! [Y: relational_fmla_a_b] :
( ( P @ Y )
=> ( Y = X6 ) ) )
=> ( P @ ( the_Re7701088767514941656la_a_b @ P ) ) ) ).
% theI'
thf(fact_684_theI,axiom,
! [P: nat > $o,A2: nat] :
( ( P @ A2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the_nat @ P ) ) ) ) ).
% theI
thf(fact_685_theI,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ( P @ A2 )
=> ( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the_Re7701088767514941656la_a_b @ P ) ) ) ) ).
% theI
thf(fact_686_the1__equality_H,axiom,
! [P: nat > $o,A2: nat] :
( ( uniq_nat @ P )
=> ( ( P @ A2 )
=> ( ( the_nat @ P )
= A2 ) ) ) ).
% the1_equality'
thf(fact_687_the1__equality_H,axiom,
! [P: relational_fmla_a_b > $o,A2: relational_fmla_a_b] :
( ( uniq_R5111430000950013146la_a_b @ P )
=> ( ( P @ A2 )
=> ( ( the_Re7701088767514941656la_a_b @ P )
= A2 ) ) ) ).
% the1_equality'
thf(fact_688_cov_H__cov,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X4 @ Q3 @ G )
=> ( relational_cov_a_b @ X4 @ Q3 @ G ) ) ).
% cov'_cov
thf(fact_689_cov__cov_H,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( relational_cov_a_b2 @ X4 @ Q3 @ G ) ) ).
% cov_cov'
thf(fact_690_cov__eq__cov_H,axiom,
relational_cov_a_b = relational_cov_a_b2 ).
% cov_eq_cov'
thf(fact_691_gen_H__gen,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X4 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ Q3 @ G ) ) ).
% gen'_gen
thf(fact_692_gen__gen_H,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( relational_gen_a_b2 @ X4 @ Q3 @ G ) ) ).
% gen_gen'
thf(fact_693_gen__eq__gen_H,axiom,
relational_gen_a_b = relational_gen_a_b2 ).
% gen_eq_gen'
thf(fact_694_gen_H__qp,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b2 @ X4 @ Q3 @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen'_qp
thf(fact_695_cov_Ononfree,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_cov_a_b @ X4 @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ).
% cov.nonfree
thf(fact_696_cov__fv,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q3 ) ) ) ) ) ) ).
% cov_fv
thf(fact_697_gen_H_Ointros_I1_J,axiom,
! [X4: nat] : ( relational_gen_a_b2 @ X4 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen'.intros(1)
thf(fact_698_cov__fv__aux,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ ( relational_fv_a_b @ Q3 ) ) ) ) ) ).
% cov_fv_aux
thf(fact_699_Greatest__def,axiom,
( order_5724808138429204845et_nat
= ( ^ [P2: set_nat > $o] :
( the_set_nat
@ ^ [X: set_nat] :
( ( P2 @ X )
& ! [Y5: set_nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_700_Greatest__def,axiom,
( order_6069613389275354556la_a_b
= ( ^ [P2: set_Re381260168593705685la_a_b > $o] :
( the_se5712841328101729806la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( P2 @ X )
& ! [Y5: set_Re381260168593705685la_a_b] :
( ( P2 @ Y5 )
=> ( ord_le4112832032246704949la_a_b @ Y5 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_701_Greatest__def,axiom,
( order_Greatest_nat_o
= ( ^ [P2: ( nat > $o ) > $o] :
( the_nat_o
@ ^ [X: nat > $o] :
( ( P2 @ X )
& ! [Y5: nat > $o] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat_o @ Y5 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_702_Greatest__def,axiom,
( order_7602139253572565961_a_b_o
= ( ^ [P2: ( relational_fmla_a_b > $o ) > $o] :
( the_Re3096307281707904503_a_b_o
@ ^ [X: relational_fmla_a_b > $o] :
( ( P2 @ X )
& ! [Y5: relational_fmla_a_b > $o] :
( ( P2 @ Y5 )
=> ( ord_le7191224889845164944_a_b_o @ Y5 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_703_Greatest__def,axiom,
( order_Greatest_nat
= ( ^ [P2: nat > $o] :
( the_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_704_Powp__mono,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ord_le3964352015994296041_nat_o @ ( powp_nat @ A ) @ ( powp_nat @ B ) ) ) ).
% Powp_mono
thf(fact_705_Powp__mono,axiom,
! [A: relational_fmla_a_b > $o,B: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A @ B )
=> ( ord_le8511361905821865178_a_b_o @ ( powp_R3848420074929365056la_a_b @ A ) @ ( powp_R3848420074929365056la_a_b @ B ) ) ) ).
% Powp_mono
thf(fact_706_flat__Disj_Osimps_I4_J,axiom,
! [V: nat,Va: relational_term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Eq_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(4)
thf(fact_707_flat__Disj_Osimps_I5_J,axiom,
! [V: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Neg_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(5)
thf(fact_708_flat__Disj_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Conj_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(6)
thf(fact_709_flat__Disj_Osimps_I7_J,axiom,
! [V: nat,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(7)
thf(fact_710_minus__apply,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B5: nat > $o,X: nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B5 @ X ) ) ) ) ).
% minus_apply
thf(fact_711_minus__apply,axiom,
( minus_9215201808853403479_a_b_o
= ( ^ [A3: relational_fmla_a_b > $o,B5: relational_fmla_a_b > $o,X: relational_fmla_a_b] : ( minus_minus_o @ ( A3 @ X ) @ ( B5 @ X ) ) ) ) ).
% minus_apply
thf(fact_712_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_713_Diff__idemp,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B ) @ B )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) ).
% Diff_idemp
thf(fact_714_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_715_Diff__iff,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
= ( ( member3481406638322139244la_a_b @ C @ A )
& ~ ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_716_Diff__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
& ~ ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_717_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_718_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_719_DiffI,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ A )
=> ( ~ ( member3481406638322139244la_a_b @ C @ B )
=> ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_720_DiffI,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( ~ ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_721_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_722_fmla_Oinject_I3_J,axiom,
! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
( ( ( relational_Eq_a_b @ X31 @ X32 )
= ( relational_Eq_a_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fmla.inject(3)
thf(fact_723_fmla_Oinject_I4_J,axiom,
! [X42: relational_fmla_a_b,Y42: relational_fmla_a_b] :
( ( ( relational_Neg_a_b @ X42 )
= ( relational_Neg_a_b @ Y42 ) )
= ( X42 = Y42 ) ) ).
% fmla.inject(4)
thf(fact_724_fmla_Oinject_I5_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
( ( ( relational_Conj_a_b @ X51 @ X52 )
= ( relational_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fmla.inject(5)
thf(fact_725_fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
= ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fmla.inject(7)
thf(fact_726_Diff__cancel,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A @ A )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_cancel
thf(fact_727_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_728_empty__Diff,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ bot_bo4495933725496725865la_a_b @ A )
= bot_bo4495933725496725865la_a_b ) ).
% empty_Diff
thf(fact_729_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_730_Diff__empty,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% Diff_empty
thf(fact_731_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_732_insert__Diff1,axiom,
! [X4: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_733_insert__Diff1,axiom,
! [X4: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ A ) @ B )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_734_insert__Diff1,axiom,
! [X4: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ B )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_735_insert__Diff1,axiom,
! [X4: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_736_Diff__insert0,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ B ) )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_737_Diff__insert0,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X4 @ B ) )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_738_Diff__insert0,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ B ) )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_739_Diff__insert0,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_740_Diff__eq__empty__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_741_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_742_insert__Diff__single,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_743_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_744_set__diff__eq,axiom,
( minus_4705846553145473764la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A3 )
& ~ ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_745_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A3 )
& ~ ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_746_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A3 )
& ~ ( member_nat_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_747_set__diff__eq,axiom,
( minus_3526376402619701533la_a_b
= ( ^ [A3: set_na7556516505497143492la_a_b,B5: set_na7556516505497143492la_a_b] :
( collec4193374254315349731la_a_b
@ ^ [X: nat > relational_fmla_a_b] :
( ( member8923333377441230501la_a_b @ X @ A3 )
& ~ ( member8923333377441230501la_a_b @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_748_set__diff__eq,axiom,
( minus_1533117980695046557_b_nat
= ( ^ [A3: set_Re5563258083572488516_b_nat,B5: set_Re5563258083572488516_b_nat] :
( collec115744212963325795_b_nat
@ ^ [X: relational_fmla_a_b > nat] :
( ( member4845703336089206565_b_nat @ X @ A3 )
& ~ ( member4845703336089206565_b_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_749_set__diff__eq,axiom,
( minus_206506763509428588la_a_b
= ( ^ [A3: set_Re1288005135514575379la_a_b,B5: set_Re1288005135514575379la_a_b] :
( collec5041345499257167282la_a_b
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
( ( member8433577210552456052la_a_b @ X @ A3 )
& ~ ( member8433577210552456052la_a_b @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_750_set__diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A3 )
& ~ ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_751_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ~ ( member_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_752_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_753_DiffD2,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ~ ( member3481406638322139244la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_754_DiffD2,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ~ ( member4680049679412964150la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_755_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_756_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_757_DiffD1,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ( member3481406638322139244la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_758_DiffD1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ( member4680049679412964150la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_759_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_760_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_761_DiffE,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ~ ( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_762_DiffE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_763_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_764_fmla_Odistinct_I23_J,axiom,
! [X31: nat,X32: relational_term_a,X42: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(23)
thf(fact_765_fmla_Odistinct_I25_J,axiom,
! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(25)
thf(fact_766_fmla_Odistinct_I29_J,axiom,
! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(29)
thf(fact_767_fmla_Odistinct_I31_J,axiom,
! [X42: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X42 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(31)
thf(fact_768_fmla_Odistinct_I35_J,axiom,
! [X42: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X42 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(35)
thf(fact_769_fmla_Odistinct_I39_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(39)
thf(fact_770_fun__diff__def,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B5: nat > $o,X: nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B5 @ X ) ) ) ) ).
% fun_diff_def
thf(fact_771_fun__diff__def,axiom,
( minus_9215201808853403479_a_b_o
= ( ^ [A3: relational_fmla_a_b > $o,B5: relational_fmla_a_b > $o,X: relational_fmla_a_b] : ( minus_minus_o @ ( A3 @ X ) @ ( B5 @ X ) ) ) ) ).
% fun_diff_def
thf(fact_772_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_773_fv_Osimps_I7_J,axiom,
! [Z: nat,Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ).
% fv.simps(7)
thf(fact_774_fmla_Odistinct_I21_J,axiom,
! [X22: $o,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(21)
thf(fact_775_fmla_Odistinct_I17_J,axiom,
! [X22: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(17)
thf(fact_776_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_777_double__diff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C2 )
=> ( ( minus_4077726661957047470la_a_b @ B @ ( minus_4077726661957047470la_a_b @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_778_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_779_Diff__subset,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_780_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_781_Diff__mono,axiom,
! [A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ C2 )
=> ( ( ord_le4112832032246704949la_a_b @ D2 @ B )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B ) @ ( minus_4077726661957047470la_a_b @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_782_fmla_Odistinct_I15_J,axiom,
! [X22: $o,X42: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(15)
thf(fact_783_fmla_Odistinct_I13_J,axiom,
! [X22: $o,X31: nat,X32: relational_term_a] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(13)
thf(fact_784_insert__Diff__if,axiom,
! [X4: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ( member_set_nat @ X4 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) )
& ( ~ ( member_set_nat @ X4 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( insert_set_nat @ X4 @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_785_insert__Diff__if,axiom,
! [X4: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( ( member3481406638322139244la_a_b @ X4 @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ A ) @ B )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X4 @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ A ) @ B )
= ( insert2023870700798818565la_a_b @ X4 @ ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_786_insert__Diff__if,axiom,
! [X4: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ B )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X4 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ B )
= ( insert7010464514620295119la_a_b @ X4 @ ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_787_insert__Diff__if,axiom,
! [X4: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_788_fv_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
= ( relational_fv_a_b @ Phi ) ) ).
% fv.simps(4)
thf(fact_789_gen_Ointros_I7_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_gen_a_b @ X4 @ Q1 @ G )
| ( relational_gen_a_b @ X4 @ Q22 @ G ) )
=> ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ).
% gen.intros(7)
thf(fact_790_Gen__Conj_I1_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ Q1 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X_1 ) ) ).
% Gen_Conj(1)
thf(fact_791_Gen__Conj_I2_J,axiom,
! [X4: nat,Q22: relational_fmla_a_b,Q1: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ Q22 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X_1 ) ) ).
% Gen_Conj(2)
thf(fact_792_gen_Ointros_I3_J,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) @ G ) ) ).
% gen.intros(3)
thf(fact_793_cov_ONeg,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( relational_cov_a_b @ X4 @ ( relational_Neg_a_b @ Q3 ) @ G ) ) ).
% cov.Neg
thf(fact_794_genempty_Ointros_I9_J,axiom,
! [Q3: relational_fmla_a_b,Y3: nat] :
( ( relati5999705594545617851ty_a_b @ Q3 )
=> ( relati5999705594545617851ty_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) ) ) ).
% genempty.intros(9)
thf(fact_795_cov_H_ONeg,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X4 @ Q3 @ G )
=> ( relational_cov_a_b2 @ X4 @ ( relational_Neg_a_b @ Q3 ) @ G ) ) ).
% cov'.Neg
thf(fact_796_gen_H_Ointros_I7_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( ( relational_gen_a_b2 @ X4 @ Q1 @ G )
| ( relational_gen_a_b2 @ X4 @ Q22 @ G ) )
=> ( relational_gen_a_b2 @ X4 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ).
% gen'.intros(7)
thf(fact_797_gen_H_Ointros_I3_J,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X4 @ Q3 @ G )
=> ( relational_gen_a_b2 @ X4 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) @ G ) ) ).
% gen'.intros(3)
thf(fact_798_genempty_Ointros_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( ( relati5999705594545617851ty_a_b @ Q1 )
| ( relati5999705594545617851ty_a_b @ Q22 ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ).
% genempty.intros(6)
thf(fact_799_genempty_Ointros_I2_J,axiom,
! [Q3: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q3 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) ) ).
% genempty.intros(2)
thf(fact_800_qp__Conj,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ).
% qp_Conj
thf(fact_801_qp__Neg,axiom,
! [Q3: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Neg_a_b @ Q3 ) ) ).
% qp_Neg
thf(fact_802_diff__shunt__var,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( ( minus_minus_nat_o @ X4 @ Y3 )
= bot_bot_nat_o )
= ( ord_less_eq_nat_o @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_803_diff__shunt__var,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( ( minus_9215201808853403479_a_b_o @ X4 @ Y3 )
= bot_bo8852203127187332700_a_b_o )
= ( ord_le7191224889845164944_a_b_o @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_804_diff__shunt__var,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ X4 @ Y3 )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_805_diff__shunt__var,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_806_subset__minus__empty,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( minus_4077726661957047470la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b ) ) ).
% subset_minus_empty
thf(fact_807_subset__minus__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% subset_minus_empty
thf(fact_808_Diff__insert,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A @ B ) @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Diff_insert
thf(fact_809_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_810_insert__Diff,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_811_insert__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( insert2023870700798818565la_a_b @ A2 @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_812_insert__Diff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_813_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_814_Diff__insert2,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_815_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_816_insert__minus__eq,axiom,
! [X4: relational_fmla_a_b,Y3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ ( insert7010464514620295119la_a_b @ Y3 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ X4 @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ Y3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_minus_eq
thf(fact_817_insert__minus__eq,axiom,
! [X4: nat,Y3: nat,A: set_nat] :
( ( X4 != Y3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_818_Diff__insert__absorb,axiom,
! [X4: set_nat,A: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_819_Diff__insert__absorb,axiom,
! [X4: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X4 @ A ) @ ( insert2023870700798818565la_a_b @ X4 @ bot_bo2891247006866115487la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_820_Diff__insert__absorb,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_821_Diff__insert__absorb,axiom,
! [X4: nat,A: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_822_set__minus__singleton__eq,axiom,
! [X4: set_nat,X5: set_set_nat] :
( ~ ( member_set_nat @ X4 @ X5 )
=> ( ( minus_2163939370556025621et_nat @ X5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_823_set__minus__singleton__eq,axiom,
! [X4: set_Re381260168593705685la_a_b,X5: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X4 @ X5 )
=> ( ( minus_4705846553145473764la_a_b @ X5 @ ( insert2023870700798818565la_a_b @ X4 @ bot_bo2891247006866115487la_a_b ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_824_set__minus__singleton__eq,axiom,
! [X4: relational_fmla_a_b,X5: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X4 @ X5 )
=> ( ( minus_4077726661957047470la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_825_set__minus__singleton__eq,axiom,
! [X4: nat,X5: set_nat] :
( ~ ( member_nat @ X4 @ X5 )
=> ( ( minus_minus_set_nat @ X5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_826_subset__Diff__insert,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat @ X4 @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ C2 ) )
& ~ ( member_set_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_827_subset__Diff__insert,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X4: set_Re381260168593705685la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ ( minus_4705846553145473764la_a_b @ B @ ( insert2023870700798818565la_a_b @ X4 @ C2 ) ) )
= ( ( ord_le1577343677690852715la_a_b @ A @ ( minus_4705846553145473764la_a_b @ B @ C2 ) )
& ~ ( member3481406638322139244la_a_b @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_828_subset__Diff__insert,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B @ ( insert7010464514620295119la_a_b @ X4 @ C2 ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B @ C2 ) )
& ~ ( member4680049679412964150la_a_b @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_829_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X4: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_830_qp__Exists,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_qp_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) ) ) ) ).
% qp_Exists
thf(fact_831_qp__ExistsE,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) )
=> ~ ( ( relational_qp_a_b @ Q3 )
=> ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) ) ) ) ).
% qp_ExistsE
thf(fact_832_sub_Osimps_I7_J,axiom,
! [Z: nat,Q3: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) ).
% sub.simps(7)
thf(fact_833_sub_Osimps_I4_J,axiom,
! [Q3: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q3 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) ).
% sub.simps(4)
thf(fact_834_Diff__single__insert,axiom,
! [A: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) @ B )
=> ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_835_Diff__single__insert,axiom,
! [A: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_836_subset__insert__iff,axiom,
! [A: set_set_nat,X4: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X4 @ B ) )
= ( ( ( member_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B ) )
& ( ~ ( member_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_837_subset__insert__iff,axiom,
! [A: set_se6865892389300016395la_a_b,X4: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ ( insert2023870700798818565la_a_b @ X4 @ B ) )
= ( ( ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ord_le1577343677690852715la_a_b @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X4 @ bot_bo2891247006866115487la_a_b ) ) @ B ) )
& ( ~ ( member3481406638322139244la_a_b @ X4 @ A )
=> ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_838_subset__insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_839_subset__insert__iff,axiom,
! [A: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_840_Compl__insert,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( uminus235785408959094814la_a_b @ ( insert7010464514620295119la_a_b @ X4 @ A ) )
= ( minus_4077726661957047470la_a_b @ ( uminus235785408959094814la_a_b @ A ) @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Compl_insert
thf(fact_841_Compl__insert,axiom,
! [X4: nat,A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ A ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_842_sub_Osimps_I3_J,axiom,
! [X4: nat,T2: relational_term_a] :
( ( relational_sub_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(3)
thf(fact_843_it__step__insert__iff,axiom,
! [It: set_set_nat,S: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ It @ S )
=> ( ( member_set_nat @ X4 @ It )
=> ( ( minus_2163939370556025621et_nat @ S @ ( minus_2163939370556025621et_nat @ It @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ X4 @ ( minus_2163939370556025621et_nat @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_844_it__step__insert__iff,axiom,
! [It: set_se6865892389300016395la_a_b,S: set_se6865892389300016395la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ It @ S )
=> ( ( member3481406638322139244la_a_b @ X4 @ It )
=> ( ( minus_4705846553145473764la_a_b @ S @ ( minus_4705846553145473764la_a_b @ It @ ( insert2023870700798818565la_a_b @ X4 @ bot_bo2891247006866115487la_a_b ) ) )
= ( insert2023870700798818565la_a_b @ X4 @ ( minus_4705846553145473764la_a_b @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_845_it__step__insert__iff,axiom,
! [It: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ It @ S )
=> ( ( member4680049679412964150la_a_b @ X4 @ It )
=> ( ( minus_4077726661957047470la_a_b @ S @ ( minus_4077726661957047470la_a_b @ It @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ X4 @ ( minus_4077726661957047470la_a_b @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_846_it__step__insert__iff,axiom,
! [It: set_nat,S: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ It @ S )
=> ( ( member_nat @ X4 @ It )
=> ( ( minus_minus_set_nat @ S @ ( minus_minus_set_nat @ It @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_847_insert__remove__id,axiom,
! [X4: set_nat,X5: set_set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( X5
= ( insert_set_nat @ X4 @ ( minus_2163939370556025621et_nat @ X5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_848_insert__remove__id,axiom,
! [X4: set_Re381260168593705685la_a_b,X5: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ X5 )
=> ( X5
= ( insert2023870700798818565la_a_b @ X4 @ ( minus_4705846553145473764la_a_b @ X5 @ ( insert2023870700798818565la_a_b @ X4 @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_849_insert__remove__id,axiom,
! [X4: relational_fmla_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ X5 )
=> ( X5
= ( insert7010464514620295119la_a_b @ X4 @ ( minus_4077726661957047470la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_850_insert__remove__id,axiom,
! [X4: nat,X5: set_nat] :
( ( member_nat @ X4 @ X5 )
=> ( X5
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ X5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_851_remove__def,axiom,
( remove4261432235257513082la_a_b
= ( ^ [X: relational_fmla_a_b,A3: set_Re381260168593705685la_a_b] : ( minus_4077726661957047470la_a_b @ A3 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% remove_def
thf(fact_852_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_853_fv__exists,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X4 @ Q3 ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Q3 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% fv_exists
thf(fact_854_fv__erase,axiom,
! [Q3: relational_fmla_a_b,X4: nat] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_erase_a_b @ Q3 @ X4 ) ) @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q3 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% fv_erase
thf(fact_855_gen__Eq,axiom,
! [Z: nat,A2: nat,T2: relational_term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ ( relational_Eq_a_b @ A2 @ T2 ) @ G )
= ( ( Z = A2 )
& ? [C4: a] :
( ( T2
= ( relational_Const_a @ C4 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A2 @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% gen_Eq
thf(fact_856_qp__impl_Osimps_I3_J,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) )
= ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) ) ).
% qp_impl.simps(3)
thf(fact_857_term_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( relational_Const_a @ X1 )
= ( relational_Const_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_858_term_Oinject_I1_J,axiom,
! [X1: nat,Y1: nat] :
( ( ( relational_Const_nat @ X1 )
= ( relational_Const_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_859_term_Oinject_I1_J,axiom,
! [X1: relational_fmla_a_b,Y1: relational_fmla_a_b] :
( ( ( relati3340789340644911146la_a_b @ X1 )
= ( relati3340789340644911146la_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_860_member__remove,axiom,
! [X4: set_nat,Y3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ ( remove_set_nat @ Y3 @ A ) )
= ( ( member_set_nat @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_861_member__remove,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ ( remove6344774115224635824la_a_b @ Y3 @ A ) )
= ( ( member3481406638322139244la_a_b @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_862_member__remove,axiom,
! [X4: relational_fmla_a_b,Y3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ ( remove4261432235257513082la_a_b @ Y3 @ A ) )
= ( ( member4680049679412964150la_a_b @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_863_member__remove,axiom,
! [X4: nat,Y3: nat,A: set_nat] :
( ( member_nat @ X4 @ ( remove_nat @ Y3 @ A ) )
= ( ( member_nat @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_864_qp__impl_Osimps_I1_J,axiom,
! [X4: nat,C: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) ) ).
% qp_impl.simps(1)
thf(fact_865_erase_Osimps_I4_J,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_erase_a_b @ ( relational_Neg_a_b @ Q3 ) @ X4 )
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q3 @ X4 ) ) ) ).
% erase.simps(4)
thf(fact_866_erase_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat] :
( ( relational_erase_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X4 )
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q1 @ X4 ) @ ( relational_erase_a_b @ Q22 @ X4 ) ) ) ).
% erase.simps(5)
thf(fact_867_erase_Osimps_I1_J,axiom,
! [T2: $o,X4: nat] :
( ( relational_erase_a_b @ ( relational_Bool_a_b @ T2 ) @ X4 )
= ( relational_Bool_a_b @ T2 ) ) ).
% erase.simps(1)
thf(fact_868_erase_Osimps_I7_J,axiom,
! [X4: nat,Z: nat,Q3: relational_fmla_a_b] :
( ( ( X4 = Z )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ X4 )
= ( relati591517084277583526ts_a_b @ X4 @ Q3 ) ) )
& ( ( X4 != Z )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ X4 )
= ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q3 @ X4 ) ) ) ) ) ).
% erase.simps(7)
thf(fact_869_Gen__erase,axiom,
! [X4: nat,Q3: relational_fmla_a_b,Z: nat] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ Q3 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_erase_a_b @ Q3 @ Z ) @ X_1 ) ) ).
% Gen_erase
thf(fact_870_gen__Gen__erase,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z: nat] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_erase_a_b @ Q3 @ Z ) @ X_1 ) ) ).
% gen_Gen_erase
thf(fact_871_exists,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_qp_a_b @ Q3 )
=> ( relational_qp_a_b @ ( relati3989891337220013914ts_a_b @ X4 @ Q3 ) ) ) ).
% exists
thf(fact_872_minus__set__def,axiom,
( minus_4705846553145473764la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ( minus_5536082475972203681_a_b_o
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ A3 )
@ ^ [X: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_873_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_874_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A3: set_nat_nat,B5: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A3 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_875_minus__set__def,axiom,
( minus_3526376402619701533la_a_b
= ( ^ [A3: set_na7556516505497143492la_a_b,B5: set_na7556516505497143492la_a_b] :
( collec4193374254315349731la_a_b
@ ( minus_5606807804648251304_a_b_o
@ ^ [X: nat > relational_fmla_a_b] : ( member8923333377441230501la_a_b @ X @ A3 )
@ ^ [X: nat > relational_fmla_a_b] : ( member8923333377441230501la_a_b @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_876_minus__set__def,axiom,
( minus_1533117980695046557_b_nat
= ( ^ [A3: set_Re5563258083572488516_b_nat,B5: set_Re5563258083572488516_b_nat] :
( collec115744212963325795_b_nat
@ ( minus_3406981994672221992_nat_o
@ ^ [X: relational_fmla_a_b > nat] : ( member4845703336089206565_b_nat @ X @ A3 )
@ ^ [X: relational_fmla_a_b > nat] : ( member4845703336089206565_b_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_877_minus__set__def,axiom,
( minus_206506763509428588la_a_b
= ( ^ [A3: set_Re1288005135514575379la_a_b,B5: set_Re1288005135514575379la_a_b] :
( collec5041345499257167282la_a_b
@ ( minus_5544186201268655641_a_b_o
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ A3 )
@ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_878_minus__set__def,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( minus_9215201808853403479_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A3 )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_879_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_880_qp__impl_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Neg_a_b @ V ) ) ).
% qp_impl.simps(6)
thf(fact_881_qp__impl_Osimps_I7_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).
% qp_impl.simps(7)
thf(fact_882_qp__impl_Osimps_I4_J,axiom,
! [V: $o] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).
% qp_impl.simps(4)
thf(fact_883_qp__code,axiom,
relational_qp_a_b = relati3725921752842749053pl_a_b ).
% qp_code
thf(fact_884_qp__imp__qp__impl,axiom,
! [Q3: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q3 )
=> ( relati3725921752842749053pl_a_b @ Q3 ) ) ).
% qp_imp_qp_impl
thf(fact_885_qp__impl__imp__qp,axiom,
! [Q3: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ Q3 )
=> ( relational_qp_a_b @ Q3 ) ) ).
% qp_impl_imp_qp
thf(fact_886_exists__def,axiom,
( relati3989891337220013914ts_a_b
= ( ^ [X: nat,Q: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ) ).
% exists_def
thf(fact_887_exists__Exists,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relati3989891337220013914ts_a_b @ X4 @ Q3 )
= ( relati591517084277583526ts_a_b @ X4 @ Q3 ) ) ) ).
% exists_Exists
thf(fact_888_Eqc,axiom,
! [X4: nat,C: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) ) ).
% Eqc
thf(fact_889_qp_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relational_qp_a_b @ A2 )
=> ( ~ ( relational_ap_a_b @ A2 )
=> ~ ! [Q6: relational_fmla_a_b] :
( ? [X3: nat] :
( A2
= ( relati3989891337220013914ts_a_b @ X3 @ Q6 ) )
=> ~ ( relational_qp_a_b @ Q6 ) ) ) ) ).
% qp.cases
thf(fact_890_qp_Osimps,axiom,
( relational_qp_a_b
= ( ^ [A5: relational_fmla_a_b] :
( ? [Q: relational_fmla_a_b] :
( ( A5 = Q )
& ( relational_ap_a_b @ Q ) )
| ? [Q: relational_fmla_a_b,X: nat] :
( ( A5
= ( relati3989891337220013914ts_a_b @ X @ Q ) )
& ( relational_qp_a_b @ Q ) ) ) ) ) ).
% qp.simps
thf(fact_891_qp__impl_Oelims_I2_J,axiom,
! [X4: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X4 )
=> ( ! [X3: nat,C5: a] :
( X4
!= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( X4
!= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) ) ) ) ) ).
% qp_impl.elims(2)
thf(fact_892_csts__term_Osimps_I2_J,axiom,
! [C: a] :
( ( relati6638259395341848997term_a @ ( relational_Const_a @ C ) )
= ( insert_a @ C @ bot_bot_set_a ) ) ).
% csts_term.simps(2)
thf(fact_893_csts__term_Osimps_I2_J,axiom,
! [C: relational_fmla_a_b] :
( ( relati1926769566493843000la_a_b @ ( relati3340789340644911146la_a_b @ C ) )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ).
% csts_term.simps(2)
thf(fact_894_csts__term_Osimps_I2_J,axiom,
! [C: nat] :
( ( relati694035416245573993rm_nat @ ( relational_Const_nat @ C ) )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ).
% csts_term.simps(2)
thf(fact_895_fmla_Oinject_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
( ( ( relational_Pred_b_a @ X11 @ X12 )
= ( relational_Pred_b_a @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fmla.inject(1)
thf(fact_896_fmla_Odistinct_I3_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(3)
thf(fact_897_fmla_Odistinct_I5_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X42: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(5)
thf(fact_898_fmla_Odistinct_I7_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(7)
thf(fact_899_sub_Osimps_I2_J,axiom,
! [P3: b,Ts2: list_R6823256787227418703term_a] :
( ( relational_sub_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(2)
thf(fact_900_flat__Disj_Osimps_I2_J,axiom,
! [V: b,Va: list_R6823256787227418703term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Pred_b_a @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(2)
thf(fact_901_gen__Pred,axiom,
! [Z: nat,P3: b,Ts2: list_R6823256787227418703term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ G )
= ( ( member_nat @ Z @ ( relati4569515538964159125_set_a @ Ts2 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen_Pred
thf(fact_902_gen_Ointros_I5_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen.intros(5)
thf(fact_903_gen_Ointros_I4_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).
% gen.intros(4)
thf(fact_904_fv_Osimps_I1_J,axiom,
! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
= ( relati4569515538964159125_set_a @ Ts2 ) ) ).
% fv.simps(1)
thf(fact_905_qp__impl_Oelims_I1_J,axiom,
! [X4: relational_fmla_a_b,Y3: $o] :
( ( ( relati3725921752842749053pl_a_b @ X4 )
= Y3 )
=> ( ( ? [X3: nat,C5: a] :
( X4
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) )
=> ~ Y3 )
=> ( ( ? [X3: b,Ts: list_R6823256787227418703term_a] :
( X4
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ Y3 )
=> ( ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( Y3
= ( ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) ) ) )
=> ( ( ? [V2: $o] :
( X4
= ( relational_Bool_a_b @ V2 ) )
=> Y3 )
=> ( ( ? [V2: nat,Vb: nat] :
( X4
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> Y3 )
=> ( ( ? [V2: relational_fmla_a_b] :
( X4
= ( relational_Neg_a_b @ V2 ) )
=> Y3 )
=> ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X4
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> Y3 )
=> ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X4
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> Y3 ) ) ) ) ) ) ) ) ) ).
% qp_impl.elims(1)
thf(fact_906_flat__Disj_Oelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X4 )
= Y3 )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( Y3
!= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q12 ) @ ( restri569617705344514291sj_a_b @ Q23 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: $o] :
( ( X4
= ( relational_Bool_a_b @ V2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ V2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.elims
thf(fact_907_sub_Oelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X4 )
= Y3 )
=> ( ! [T4: $o] :
( ( X4
= ( relational_Bool_a_b @ T4 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T4 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q6 ) @ ( relational_sub_a_b @ Q6 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) ) )
=> ~ ! [Z5: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ Z5 @ Q6 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z5 @ Q6 ) @ ( relational_sub_a_b @ Q6 ) ) ) ) ) ) ) ) ) ) ) ).
% sub.elims
thf(fact_908_qp__impl_Oelims_I3_J,axiom,
! [X4: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X4 )
=> ( ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) )
=> ( ! [V2: $o] :
( X4
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb: nat] :
( X4
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X4
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X4
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X4
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% qp_impl.elims(3)
thf(fact_909_Un__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
| ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_910_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_911_UnCI,axiom,
! [C: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ A ) )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_912_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_913_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_914_le__sup__iff,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X4 @ Z )
& ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_915_sup__bot_Oright__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_916_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_917_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ A2 @ B2 ) )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_918_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_919_sup__bot_Oleft__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_920_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_921_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B2 )
= bot_bo4495933725496725865la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_922_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_923_sup__eq__bot__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X4 @ Y3 )
= bot_bo4495933725496725865la_a_b )
= ( ( X4 = bot_bo4495933725496725865la_a_b )
& ( Y3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_924_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_925_bot__eq__sup__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ X4 @ Y3 ) )
= ( ( X4 = bot_bo4495933725496725865la_a_b )
& ( Y3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_926_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_927_sup__bot__right,axiom,
! [X4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X4 @ bot_bo4495933725496725865la_a_b )
= X4 ) ).
% sup_bot_right
thf(fact_928_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_929_sup__bot__left,axiom,
! [X4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_930_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_931_Un__empty,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% Un_empty
thf(fact_932_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_933_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_934_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_935_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_936_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_937_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_938_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_939_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_940_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_941_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_942_sup__absorb2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_943_sup__absorb1,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_944_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_945_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_946_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y3: nat] :
( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: nat,Y: nat,Z5: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z5 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y @ Z5 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_947_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_948_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_949_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y5: nat] :
( ( sup_sup_nat @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_950_sup__least,axiom,
! [Y3: nat,X4: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_951_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_952_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_953_le__supI2,axiom,
! [X4: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_954_le__supI1,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_955_sup__ge2,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_956_sup__ge1,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_957_le__supI,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_958_le__supE,axiom,
! [A2: nat,B2: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X4 )
=> ~ ( ord_less_eq_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_959_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_960_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_961_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X4 @ bot_bo4495933725496725865la_a_b )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_962_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_963_UnI2,axiom,
! [C: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_964_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_965_UnI1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_966_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_967_UnE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) )
=> ( ~ ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_968_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_969_Un__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A3 )
| ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_970_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
| ( member_nat @ X @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_971_Collect__disj__eq,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
| ( Q3 @ X ) ) )
= ( sup_su5130108678486352897la_a_b @ ( collec3419995626248312948la_a_b @ P ) @ ( collec3419995626248312948la_a_b @ Q3 ) ) ) ).
% Collect_disj_eq
thf(fact_972_Collect__disj__eq,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q3 @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_disj_eq
thf(fact_973_Un__empty__right,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% Un_empty_right
thf(fact_974_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_975_Un__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_976_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_977_insert__def,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A5: relational_fmla_a_b] :
( sup_su5130108678486352897la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( X = A5 ) ) ) ) ) ).
% insert_def
thf(fact_978_insert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X: nat] : ( X = A5 ) ) ) ) ) ).
% insert_def
thf(fact_979_Collect__imp__eq,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
=> ( Q3 @ X ) ) )
= ( sup_su5130108678486352897la_a_b @ ( uminus235785408959094814la_a_b @ ( collec3419995626248312948la_a_b @ P ) ) @ ( collec3419995626248312948la_a_b @ Q3 ) ) ) ).
% Collect_imp_eq
thf(fact_980_Collect__imp__eq,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
=> ( Q3 @ X ) ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_imp_eq
thf(fact_981_insert__is__Un,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A5: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A5 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_982_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_983_Un__singleton__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X4: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B )
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_984_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_985_singleton__Un__iff,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A @ B ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_986_singleton__Un__iff,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X4 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_987_gen_Ointros_I6_J,axiom,
! [X4: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q1 @ G1 )
=> ( ( relational_gen_a_b @ X4 @ Q22 @ G22 )
=> ( relational_gen_a_b @ X4 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen.intros(6)
thf(fact_988_csts__term_Osimps_I1_J,axiom,
! [X4: nat] :
( ( relati1926769566493843000la_a_b @ ( relati2528351936964124430la_a_b @ X4 ) )
= bot_bo4495933725496725865la_a_b ) ).
% csts_term.simps(1)
thf(fact_989_csts__term_Osimps_I1_J,axiom,
! [X4: nat] :
( ( relati694035416245573993rm_nat @ ( relational_Var_nat @ X4 ) )
= bot_bot_set_nat ) ).
% csts_term.simps(1)
thf(fact_990_cov_OEq__self,axiom,
! [X4: nat] : ( relational_cov_a_b @ X4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ X4 ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov.Eq_self
thf(fact_991_cov_H_OEq__self,axiom,
! [X4: nat] : ( relational_cov_a_b2 @ X4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ X4 ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov'.Eq_self
thf(fact_992_cov_OEqR,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( relational_cov_a_b @ X4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqR
thf(fact_993_cov_OEqL,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( relational_cov_a_b @ X4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqL
thf(fact_994_cov_H_OEqR,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( relational_cov_a_b2 @ X4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqR
thf(fact_995_cov_H_OEqL,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( relational_cov_a_b2 @ X4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqL
thf(fact_996_csts__term_Oelims,axiom,
! [X4: relati112041753218324778la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relati1926769566493843000la_a_b @ X4 )
= Y3 )
=> ( ( ? [X3: nat] :
( X4
= ( relati2528351936964124430la_a_b @ X3 ) )
=> ( Y3 != bot_bo4495933725496725865la_a_b ) )
=> ~ ! [C5: relational_fmla_a_b] :
( ( X4
= ( relati3340789340644911146la_a_b @ C5 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ C5 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% csts_term.elims
thf(fact_997_csts__term_Oelims,axiom,
! [X4: relational_term_nat,Y3: set_nat] :
( ( ( relati694035416245573993rm_nat @ X4 )
= Y3 )
=> ( ( ? [X3: nat] :
( X4
= ( relational_Var_nat @ X3 ) )
=> ( Y3 != bot_bot_set_nat ) )
=> ~ ! [C5: nat] :
( ( X4
= ( relational_Const_nat @ C5 ) )
=> ( Y3
!= ( insert_nat @ C5 @ bot_bot_set_nat ) ) ) ) ) ).
% csts_term.elims
thf(fact_998_eqs__minus,axiom,
! [X4: nat,G: set_Re381260168593705685la_a_b,Y3: nat] :
( ( relational_eqs_a_b @ X4 @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
= ( minus_minus_set_nat @ ( relational_eqs_a_b @ X4 @ G ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) ).
% eqs_minus
thf(fact_999_nocp_Oelims_I1_J,axiom,
! [X4: relational_fmla_a_b,Y3: $o] :
( ( ( relational_nocp_a_b @ X4 )
= Y3 )
=> ( ( ? [B3: $o] :
( X4
= ( relational_Bool_a_b @ B3 ) )
=> Y3 )
=> ( ( ? [P4: b,Ts: list_R6823256787227418703term_a] :
( X4
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ Y3 )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( Y3
= ( T4
= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( Y3
= ( ~ ( relational_nocp_a_b @ Q6 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( Y3
= ( ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( Y3
= ( ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( Y3
= ( ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.elims(1)
thf(fact_1000_nocp_Oelims_I2_J,axiom,
! [X4: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X4 )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( X4
!= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( T4
= ( relational_Var_a @ X3 ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ~ ( relational_nocp_a_b @ Q6 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(2)
thf(fact_1001_nocp_Oelims_I3_J,axiom,
! [X4: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X4 )
=> ( ! [B3: $o] :
( X4
!= ( relational_Bool_a_b @ B3 ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( T4
!= ( relational_Var_a @ X3 ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( relational_nocp_a_b @ Q6 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(3)
thf(fact_1002_gen__eqs,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z: nat] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( relational_eqs_a_b @ Z @ G )
= bot_bot_set_nat ) ) ).
% gen_eqs
thf(fact_1003_eqs__empty,axiom,
! [X4: nat] :
( ( relational_eqs_a_b @ X4 @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_nat ) ).
% eqs_empty
thf(fact_1004_sup__Un__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su1471977682094119364_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ R )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ S ) )
= ( ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1005_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1006_sup__set__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( sup_su1471977682094119364_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A3 )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1007_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1008_nocp_Osimps_I7_J,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) )
= ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ).
% nocp.simps(7)
thf(fact_1009_fv_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(6)
thf(fact_1010_fv_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(5)
thf(fact_1011_fv_Osimps_I3_J,axiom,
! [X4: nat,T3: relational_term_a] :
( ( relational_fv_a_b @ ( relational_Eq_a_b @ X4 @ T3 ) )
= ( sup_sup_set_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T3 ) ) ) ).
% fv.simps(3)
thf(fact_1012_notin__eqs,axiom,
! [X4: nat,Y3: nat,G: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ G )
=> ~ ( member_nat @ Y3 @ ( relational_eqs_a_b @ X4 @ G ) ) ) ).
% notin_eqs
thf(fact_1013_eqs__in,axiom,
! [Y3: nat,X4: nat,G: set_Re381260168593705685la_a_b] :
( ( member_nat @ Y3 @ ( relational_eqs_a_b @ X4 @ G ) )
=> ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ G ) ) ).
% eqs_in
thf(fact_1014_eqs__def,axiom,
( relational_eqs_a_b
= ( ^ [X: nat,G4: set_Re381260168593705685la_a_b] :
( collect_nat
@ ^ [Y5: nat] :
( ( X != Y5 )
& ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y5 ) ) @ G4 ) ) ) ) ) ).
% eqs_def
thf(fact_1015_fv_Oelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_nat] :
( ( ( relational_fv_a_b @ X4 )
= Y3 )
=> ( ! [Uu3: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ Uu3 @ Ts ) )
=> ( Y3
!= ( relati4569515538964159125_set_a @ Ts ) ) )
=> ( ( ? [B3: $o] :
( X4
= ( relational_Bool_a_b @ B3 ) )
=> ( Y3 != bot_bot_set_nat ) )
=> ( ! [X3: nat,T5: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T5 ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Phi2 ) )
=> ( Y3
!= ( relational_fv_a_b @ Phi2 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ~ ! [Z5: nat,Phi2: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ Z5 @ Phi2 ) )
=> ( Y3
!= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z5 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).
% fv.elims
thf(fact_1016_qps__minus,axiom,
! [G: set_Re381260168593705685la_a_b,X4: nat,Y3: nat] :
( ( relational_qps_a_b @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
= ( relational_qps_a_b @ G ) ) ).
% qps_minus
thf(fact_1017_fv_Opelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_nat] :
( ( ( relational_fv_a_b @ X4 )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X4 )
=> ( ! [Uu3: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ Uu3 @ Ts ) )
=> ( ( Y3
= ( relati4569515538964159125_set_a @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu3 @ Ts ) ) ) )
=> ( ! [B3: $o] :
( ( X4
= ( relational_Bool_a_b @ B3 ) )
=> ( ( Y3 = bot_bot_set_nat )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B3 ) ) ) )
=> ( ! [X3: nat,T5: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T5 ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X3 @ T5 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Phi2 ) )
=> ( ( Y3
= ( relational_fv_a_b @ Phi2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ~ ! [Z5: nat,Phi2: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ Z5 @ Phi2 ) )
=> ( ( Y3
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z5 @ bot_bot_set_nat ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z5 @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv.pelims
thf(fact_1018_gen__qps,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( relational_qps_a_b @ G )
= G ) ) ).
% gen_qps
thf(fact_1019_qps__empty,axiom,
( ( relational_qps_a_b @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% qps_empty
thf(fact_1020_qps__in,axiom,
! [Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q3 @ ( relational_qps_a_b @ G ) )
=> ( member4680049679412964150la_a_b @ Q3 @ G ) ) ).
% qps_in
thf(fact_1021_qps__qp,axiom,
! [Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q3 @ ( relational_qps_a_b @ G ) )
=> ( relational_qp_a_b @ Q3 ) ) ).
% qps_qp
thf(fact_1022_qps__def,axiom,
( relational_qps_a_b
= ( ^ [G4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q @ G4 )
& ( relational_qp_a_b @ Q ) ) ) ) ) ).
% qps_def
thf(fact_1023_flat__Disj_Opelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X4 )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ X4 )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( Y3
= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q12 ) @ ( restri569617705344514291sj_a_b @ Q23 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [V2: $o] :
( ( X4
= ( relational_Bool_a_b @ V2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ V2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.pelims
thf(fact_1024_sub_Opelims,axiom,
! [X4: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X4 )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X4 )
=> ( ! [T4: $o] :
( ( X4
= ( relational_Bool_a_b @ T4 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T4 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T4 ) ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q6 ) @ ( relational_sub_a_b @ Q6 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q6 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q23 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) ) )
=> ~ ! [Z5: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ Z5 @ Q6 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z5 @ Q6 ) @ ( relational_sub_a_b @ Q6 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z5 @ Q6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sub.pelims
thf(fact_1025_qp__impl_Opelims_I1_J,axiom,
! [X4: relational_fmla_a_b,Y3: $o] :
( ( ( relati3725921752842749053pl_a_b @ X4 )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X4 )
=> ( ! [X3: nat,C5: a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X3 @ Ts ) ) ) )
=> ( ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( Y3
= ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) ) ) )
=> ( ! [V2: $o] :
( ( X4
= ( relational_Bool_a_b @ V2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Vb: nat] :
( ( X4
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ V2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(1)
thf(fact_1026_qp__impl_Opelims_I3_J,axiom,
! [X4: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X4 )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X4 )
=> ( ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) ) )
=> ( ! [V2: $o] :
( ( X4
= ( relational_Bool_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
=> ( ! [V2: nat,Vb: nat] :
( ( X4
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(3)
thf(fact_1027_qp__impl_Opelims_I2_J,axiom,
! [X4: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X4 )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X4 )
=> ( ! [X3: nat,C5: a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C5 ) ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X3 @ Ts ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_qp_a_b @ Q6 ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(2)
thf(fact_1028_nocp_Opelims_I1_J,axiom,
! [X4: relational_fmla_a_b,Y3: $o] :
( ( ( relational_nocp_a_b @ X4 )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X4 )
=> ( ! [B3: $o] :
( ( X4
= ( relational_Bool_a_b @ B3 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B3 ) ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( ( Y3
= ( T4
!= ( relational_Var_a @ X3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( ( Y3
= ( relational_nocp_a_b @ Q6 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q6 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( Y3
= ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( Y3
= ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( Y3
= ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(1)
thf(fact_1029_nocp_Opelims_I2_J,axiom,
! [X4: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X4 )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X4 )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X4
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( T4
= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q6 ) )
=> ~ ( relational_nocp_a_b @ Q6 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(2)
thf(fact_1030_nocp_Opelims_I3_J,axiom,
! [X4: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X4 )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X4 )
=> ( ! [B3: $o] :
( ( X4
= ( relational_Bool_a_b @ B3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B3 ) ) )
=> ( ! [X3: nat,T4: relational_term_a] :
( ( X4
= ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T4 ) )
=> ( T4
!= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( X4
= ( relational_Neg_a_b @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q6 ) )
=> ( relational_nocp_a_b @ Q6 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( X4
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q23 ) ) ) )
=> ~ ! [X3: nat,Q6: relational_fmla_a_b] :
( ( X4
= ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
& ( relational_nocp_a_b @ Q6 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(3)
thf(fact_1031_csts__term_Opelims,axiom,
! [X4: relati112041753218324778la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relati1926769566493843000la_a_b @ X4 )
= Y3 )
=> ( ( accp_R2749021782369031905la_a_b @ relati4267809275444381249la_a_b @ X4 )
=> ( ! [X3: nat] :
( ( X4
= ( relati2528351936964124430la_a_b @ X3 ) )
=> ( ( Y3 = bot_bo4495933725496725865la_a_b )
=> ~ ( accp_R2749021782369031905la_a_b @ relati4267809275444381249la_a_b @ ( relati2528351936964124430la_a_b @ X3 ) ) ) )
=> ~ ! [C5: relational_fmla_a_b] :
( ( X4
= ( relati3340789340644911146la_a_b @ C5 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ C5 @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R2749021782369031905la_a_b @ relati4267809275444381249la_a_b @ ( relati3340789340644911146la_a_b @ C5 ) ) ) ) ) ) ) ).
% csts_term.pelims
thf(fact_1032_csts__term_Opelims,axiom,
! [X4: relational_term_nat,Y3: set_nat] :
( ( ( relati694035416245573993rm_nat @ X4 )
= Y3 )
=> ( ( accp_R1321120385092908946rm_nat @ relati5935188524400849522el_nat @ X4 )
=> ( ! [X3: nat] :
( ( X4
= ( relational_Var_nat @ X3 ) )
=> ( ( Y3 = bot_bot_set_nat )
=> ~ ( accp_R1321120385092908946rm_nat @ relati5935188524400849522el_nat @ ( relational_Var_nat @ X3 ) ) ) )
=> ~ ! [C5: nat] :
( ( X4
= ( relational_Const_nat @ C5 ) )
=> ( ( Y3
= ( insert_nat @ C5 @ bot_bot_set_nat ) )
=> ~ ( accp_R1321120385092908946rm_nat @ relati5935188524400849522el_nat @ ( relational_Const_nat @ C5 ) ) ) ) ) ) ) ).
% csts_term.pelims
thf(fact_1033_gen_H_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ Q6 @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b2 @ A1 @ Q12 @ G12 )
=> ~ ( relational_gen_a_b2 @ A1 @ Q23 @ G23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( A32 = G5 )
=> ~ ( ( relational_gen_a_b2 @ A1 @ Q12 @ G5 )
| ( relational_gen_a_b2 @ A1 @ Q23 @ G5 ) ) ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ A1 )
@ G5 ) )
=> ~ ( relational_gen_a_b2 @ Y @ Q6 @ G5 ) ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ A1 )
@ G5 ) )
=> ~ ( relational_gen_a_b2 @ Y @ Q6 @ G5 ) ) )
=> ~ ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y @ Q6 ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G5 ) )
=> ( ( A1 != Y )
=> ~ ( relational_gen_a_b2 @ A1 @ Q6 @ G5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen'.cases
thf(fact_1034_gen_H_Osimps,axiom,
( relational_gen_a_b2
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) )
& ( relational_gen_a_b2 @ A12 @ Q @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b2 @ A12 @ Q13 @ G13 )
& ( relational_gen_a_b2 @ A12 @ Q24 @ G24 ) ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
& ( ( relational_gen_a_b2 @ A12 @ Q13 @ A33 )
| ( relational_gen_a_b2 @ A12 @ Q24 @ A33 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y5 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ A12 )
@ G4 ) )
& ( relational_gen_a_b2 @ Y5 @ Q @ G4 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ A12 )
@ G4 ) )
& ( relational_gen_a_b2 @ Y5 @ Q @ G4 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y5 @ Q ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G4 ) )
& ( A12 != Y5 )
& ( relational_gen_a_b2 @ A12 @ Q @ G4 ) ) ) ) ) ) ).
% gen'.simps
thf(fact_1035_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1036_image__eqI,axiom,
! [B2: nat,F: relational_fmla_a_b > nat,X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X4 ) )
=> ( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1037_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: nat > relational_fmla_a_b,X4: nat,A: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1038_image__eqI,axiom,
! [B2: nat,F: nat > nat,X4: nat,A: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1039_image__is__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ A )
= bot_bo4495933725496725865la_a_b )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_1040_image__is__empty,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat] :
( ( ( image_4386371547000553590la_a_b @ F @ A )
= bot_bo4495933725496725865la_a_b )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1041_image__is__empty,axiom,
! [F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( ( image_341122591648980342_b_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_1042_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1043_empty__is__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( image_6790371041703824709la_a_b @ F @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_1044_empty__is__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat] :
( ( bot_bo4495933725496725865la_a_b
= ( image_4386371547000553590la_a_b @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1045_empty__is__image,axiom,
! [F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( bot_bot_set_nat
= ( image_341122591648980342_b_nat @ F @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_1046_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1047_image__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_1048_image__empty,axiom,
! [F: relational_fmla_a_b > nat] :
( ( image_341122591648980342_b_nat @ F @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1049_image__empty,axiom,
! [F: nat > relational_fmla_a_b] :
( ( image_4386371547000553590la_a_b @ F @ bot_bot_set_nat )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_1050_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1051_Setcompr__eq__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( member4680049679412964150la_a_b @ X @ A ) ) )
= ( image_6790371041703824709la_a_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1052_Setcompr__eq__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( member_nat @ X @ A ) ) )
= ( image_4386371547000553590la_a_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1053_Setcompr__eq__image,axiom,
! [F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( member4680049679412964150la_a_b @ X @ A ) ) )
= ( image_341122591648980342_b_nat @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1054_Setcompr__eq__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( member_nat @ X @ A ) ) )
= ( image_nat_nat @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1055_setcompr__eq__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_6790371041703824709la_a_b @ F @ ( collec3419995626248312948la_a_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1056_setcompr__eq__image,axiom,
! [F: nat > relational_fmla_a_b,P: nat > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_4386371547000553590la_a_b @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1057_setcompr__eq__image,axiom,
! [F: relational_fmla_a_b > nat,P: relational_fmla_a_b > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_341122591648980342_b_nat @ F @ ( collec3419995626248312948la_a_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1058_setcompr__eq__image,axiom,
! [F: nat > nat,P: nat > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1059_rev__image__eqI,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1060_rev__image__eqI,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: nat,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1061_rev__image__eqI,axiom,
! [X4: nat,A: set_nat,B2: relational_fmla_a_b,F: nat > relational_fmla_a_b] :
( ( member_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1062_rev__image__eqI,axiom,
! [X4: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1063_imageI,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X4 ) @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1064_imageI,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1065_imageI,axiom,
! [X4: nat,A: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat @ X4 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X4 ) @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1066_imageI,axiom,
! [X4: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1067_imageE,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A ) ) ) ).
% imageE
thf(fact_1068_imageE,axiom,
! [B2: relational_fmla_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1069_imageE,axiom,
! [B2: nat,F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A ) ) ) ).
% imageE
thf(fact_1070_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1071_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( image_6790371041703824709la_a_b @ F @ A ) )
& ( P @ X ) ) )
= ( image_6790371041703824709la_a_b @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1072_Compr__image__eq,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( image_4386371547000553590la_a_b @ F @ A ) )
& ( P @ X ) ) )
= ( image_4386371547000553590la_a_b @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1073_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_341122591648980342_b_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_341122591648980342_b_nat @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1074_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1075_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1076_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,B: set_nat] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_341122591648980342_b_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1077_image__subsetI,axiom,
! [A: set_nat,F: nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1078_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1079_image__Collect__subsetI,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( collec3419995626248312948la_a_b @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1080_image__Collect__subsetI,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > nat,B: set_nat] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_341122591648980342_b_nat @ F @ ( collec3419995626248312948la_a_b @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1081_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1082_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1083_gen__nocp__intros_I2_J,axiom,
! [Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X4: nat] :
( ( relational_gen_a_b @ Y3 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_subst_a_b @ Q @ Y3 @ X4 )
@ G ) ) ) ).
% gen_nocp_intros(2)
thf(fact_1084_gen__nocp__intros_I1_J,axiom,
! [Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X4: nat] :
( ( relational_gen_a_b @ Y3 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_subst_a_b @ Q @ Y3 @ X4 )
@ G ) ) ) ).
% gen_nocp_intros(1)
thf(fact_1085_image__constant__conv,axiom,
! [A: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ( A = bot_bo4495933725496725865la_a_b )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X: relational_fmla_a_b] : C
@ A )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X: relational_fmla_a_b] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_1086_image__constant__conv,axiom,
! [A: set_Re381260168593705685la_a_b,C: nat] :
( ( ( A = bot_bo4495933725496725865la_a_b )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C
@ A )
= bot_bot_set_nat ) )
& ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1087_image__constant__conv,axiom,
! [A: set_nat,C: relational_fmla_a_b] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C
@ A )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_1088_image__constant__conv,axiom,
! [A: set_nat,C: nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X: nat] : C
@ A )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X: nat] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1089_image__constant,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X: relational_fmla_a_b] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_1090_image__constant,axiom,
! [X4: nat,A: set_nat,C: relational_fmla_a_b] :
( ( member_nat @ X4 @ A )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_1091_image__constant,axiom,
! [X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,C: nat] :
( ( member4680049679412964150la_a_b @ X4 @ A )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1092_image__constant,axiom,
! [X4: nat,A: set_nat,C: nat] :
( ( member_nat @ X4 @ A )
=> ( ( image_nat_nat
@ ^ [X: nat] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1093_gen_Ointros_I10_J,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G ) ) ) ) ).
% gen.intros(10)
thf(fact_1094_Eq__eq__subst__iff,axiom,
! [Y3: nat,Z: nat,Q3: relational_fmla_a_b,X4: nat] :
( ( ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z ) )
= ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) )
= ( ( ( Z = X4 )
=> ( ( X4 = Y3 )
& ( Q3
= ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ X4 ) ) ) ) )
& ( ( Z != X4 )
=> ( ( Q3
= ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Z ) ) )
| ( Q3
= ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z ) ) )
| ( ( Z = Y3 )
& ( member4680049679412964150la_a_b @ Q3 @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ X4 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Y3 ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ).
% Eq_eq_subst_iff
thf(fact_1095_fv__subst,axiom,
! [X4: nat,Q3: relational_fmla_a_b,Y3: nat] :
( ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) )
= ( insert_nat @ Y3 @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q3 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) )
& ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) )
= ( relational_fv_a_b @ Q3 ) ) ) ) ).
% fv_subst
thf(fact_1096_cov_OExists,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ G )
=> ( relational_cov_a_b @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G ) ) ) ) ) ).
% cov.Exists
thf(fact_1097_cov_H_OExists,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b2 @ X4 @ Q3 @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ G )
=> ( relational_cov_a_b2 @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G ) ) ) ) ) ).
% cov'.Exists
thf(fact_1098_cov__nocp__intros,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q3 @ Gy )
=> ( relational_cov_a_b @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_subst_a_b @ Q @ Y3 @ X4 )
@ Gy ) ) ) ) ) ) ).
% cov_nocp_intros
thf(fact_1099_cov_H_OExists__gen,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b2 @ X4 @ Q3 @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q3 @ Gy )
=> ( relational_cov_a_b2 @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_subst_a_b @ Q @ Y3 @ X4 )
@ Gy ) ) ) ) ) ) ).
% cov'.Exists_gen
thf(fact_1100_gen__induct,axiom,
! [X1: nat,X22: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
( ( relational_gen_a_b @ X1 @ X22 @ X33 )
=> ( ! [X3: nat] : ( P @ X3 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
=> ( ! [Q6: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q6 )
=> ! [X3: nat] :
( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
=> ( P @ X3 @ Q6 @ ( insert7010464514620295119la_a_b @ Q6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ! [X3: nat,Q6: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q6 @ G5 )
=> ( ( P @ X3 @ Q6 @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G5 )
=> ( ( P @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q23: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G5 )
=> ( ( P @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q12 @ G12 )
=> ( ( P @ X3 @ Q12 @ G12 )
=> ! [Q23: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q23 @ G23 )
=> ( ( P @ X3 @ Q23 @ G23 )
=> ( P @ X3 @ ( relational_Disj_a_b @ Q12 @ Q23 ) @ ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( ( ( relational_gen_a_b @ X3 @ Q12 @ G5 )
& ( P @ X3 @ Q12 @ G5 ) )
| ( ( relational_gen_a_b @ X3 @ Q23 @ G5 )
& ( P @ X3 @ Q23 @ G5 ) ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ Q23 ) @ G5 ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y @ Q6 @ G5 )
=> ( ( P @ Y @ Q6 @ G5 )
=> ! [X3: nat] :
( P @ X3 @ ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ X3 )
@ G5 ) ) ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y @ Q6 @ G5 )
=> ( ( P @ Y @ Q6 @ G5 )
=> ! [X3: nat] :
( P @ X3 @ ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ X3 )
@ G5 ) ) ) )
=> ( ! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ! [Q6: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q6 @ G5 )
=> ( ( P @ X3 @ Q6 @ G5 )
=> ( P @ X3 @ ( relati591517084277583526ts_a_b @ Y @ Q6 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G5 ) ) ) ) )
=> ( P @ X1 @ X22 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_induct
thf(fact_1101_gen__Gen__cp,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_cp_a_b @ Q3 ) @ X_1 ) ) ).
% gen_Gen_cp
thf(fact_1102_Gen__cp,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ Q3 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X4 @ ( relational_cp_a_b @ Q3 ) @ X_1 ) ) ).
% Gen_cp
thf(fact_1103_Gen__cp__subst,axiom,
! [Z: nat,Q3: relational_fmla_a_b,X4: nat,Y3: nat] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z @ Q3 @ X_12 )
=> ( ( Z != X4 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) ) @ X_1 ) ) ) ).
% Gen_cp_subst
thf(fact_1104_fv__cp,axiom,
! [Q3: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q3 ) ) @ ( relational_fv_a_b @ Q3 ) ) ).
% fv_cp
thf(fact_1105_nongens__cp,axiom,
! [Q3: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relati62690040636126068ns_a_b @ ( relational_cp_a_b @ Q3 ) ) @ ( relati62690040636126068ns_a_b @ Q3 ) ) ).
% nongens_cp
thf(fact_1106_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X4: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ! [C6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C6 @ B )
=> ~ ( member4680049679412964150la_a_b @ X4 @ C6 ) )
=> ( ( member3481406638322139244la_a_b @ A @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X4 ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X4 @ A )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X4 @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1107_in__image__insert__iff,axiom,
! [B: set_set_nat,X4: nat,A: set_nat] :
( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ B )
=> ~ ( member_nat @ X4 @ C6 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X4 ) @ B ) )
= ( ( member_nat @ X4 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1108_gen__empty__cp,axiom,
! [Z: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z @ Q3 @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( ( relational_cp_a_b @ Q3 )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_empty_cp
thf(fact_1109_gen__cp__erase,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Qqp @ X4 ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_cp_erase
thf(fact_1110_qp__cp__erase,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_qp_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q3 @ X4 ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% qp_cp_erase
thf(fact_1111_ap__cp__erase,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_ap_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q3 @ X4 ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% ap_cp_erase
thf(fact_1112_gen_Ointros_I8_J,axiom,
! [Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X4: nat] :
( ( relational_gen_a_b @ Y3 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ Y3 @ X4 ) )
@ G ) ) ) ).
% gen.intros(8)
thf(fact_1113_gen_Ointros_I9_J,axiom,
! [Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X4: nat] :
( ( relational_gen_a_b @ Y3 @ Q3 @ G )
=> ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ Y3 @ X4 ) )
@ G ) ) ) ).
% gen.intros(9)
thf(fact_1114_cov_OExists__gen,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b @ X4 @ Q3 @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q3 @ Gy )
=> ( relational_cov_a_b @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ Y3 @ X4 ) )
@ Gy ) ) ) ) ) ) ).
% cov.Exists_gen
thf(fact_1115_cov_H__cp__intros,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X4 != Y3 )
=> ( ( relational_cov_a_b2 @ X4 @ Q3 @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q3 @ Gy )
=> ( relational_cov_a_b2 @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ Y3 @ X4 ) )
@ Gy ) ) ) ) ) ) ).
% cov'_cp_intros
thf(fact_1116_gen_Osimps,axiom,
( relational_gen_a_b
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) )
& ( relational_gen_a_b @ A12 @ Q @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q13 @ Q24 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b @ A12 @ Q13 @ G13 )
& ( relational_gen_a_b @ A12 @ Q24 @ G24 ) ) )
| ? [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q13 @ Q24 ) )
& ( ( relational_gen_a_b @ A12 @ Q13 @ A33 )
| ( relational_gen_a_b @ A12 @ Q24 @ A33 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y5 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A12 ) )
@ G4 ) )
& ( relational_gen_a_b @ Y5 @ Q @ G4 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A12 ) )
@ G4 ) )
& ( relational_gen_a_b @ Y5 @ Q @ G4 ) ) )
| ? [Y5: nat,Q: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y5 @ Q ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G4 ) )
& ( A12 != Y5 )
& ( relational_gen_a_b @ A12 @ Q @ G4 ) ) ) ) ) ) ).
% gen.simps
thf(fact_1117_gen_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ Q6 @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
=> ( ! [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q12 @ Q23 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b @ A1 @ Q12 @ G12 )
=> ~ ( relational_gen_a_b @ A1 @ Q23 @ G23 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q12 @ Q23 ) )
=> ( ( A32 = G5 )
=> ~ ( ( relational_gen_a_b @ A1 @ Q12 @ G5 )
| ( relational_gen_a_b @ A1 @ Q23 @ G5 ) ) ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y @ Q6 @ G5 ) ) )
=> ( ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y @ Q6 @ G5 ) ) )
=> ~ ! [Y: nat,Q6: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y @ Q6 ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G5 ) )
=> ( ( A1 != Y )
=> ~ ( relational_gen_a_b @ A1 @ Q6 @ G5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen.cases
thf(fact_1118_gen__sat,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( ( relational_sat_a_b @ Q3 @ I @ Sigma )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ G )
& ( relational_sat_a_b @ X3 @ I @ Sigma ) ) ) ) ).
% gen_sat
thf(fact_1119_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N: nat] :
( ( member_nat @ N @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_1120_fresh2_I3_J,axiom,
! [X4: nat,Y3: nat,Q3: relational_fmla_a_b] :
~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q3 ) @ ( relational_fv_a_b @ Q3 ) ) ).
% fresh2(3)
thf(fact_1121_gen__sat__erase,axiom,
! [Y3: nat,Q3: relational_fmla_a_b,Gy: set_Re381260168593705685la_a_b,X4: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ Y3 @ Q3 @ Gy )
=> ( ( relational_sat_a_b @ ( relational_erase_a_b @ Q3 @ X4 ) @ I @ Sigma )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ Gy )
& ( relational_sat_a_b @ X3 @ I @ Sigma ) ) ) ) ).
% gen_sat_erase
thf(fact_1122_Exists__nonfree__equiv,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) @ Q3 ) ) ).
% Exists_nonfree_equiv
thf(fact_1123_subst__exists,axiom,
! [Z: nat,Q3: relational_fmla_a_b,X4: nat,Y3: nat] :
( ( ( member_nat @ Z @ ( relational_fv_a_b @ Q3 ) )
=> ( ( ( X4 = Z )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q3 ) @ X4 @ Y3 )
= ( relati3989891337220013914ts_a_b @ X4 @ Q3 ) ) )
& ( ( X4 != Z )
=> ( ( ( Z = Y3 )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q3 ) @ X4 @ Y3 )
= ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q3 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q3 @ Z @ ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q3 ) ) @ X4 @ Y3 ) ) ) )
& ( ( Z != Y3 )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q3 ) @ X4 @ Y3 )
= ( relati3989891337220013914ts_a_b @ Z @ ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) ) ) ) ) ) ) )
& ( ~ ( member_nat @ Z @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q3 ) @ X4 @ Y3 )
= ( relational_subst_a_b @ Q3 @ X4 @ Y3 ) ) ) ) ).
% subst_exists
thf(fact_1124_rrb__simps_I7_J,axiom,
! [Y3: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Qy ) )
= ( ? [X2: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y3 @ Qy @ X2 )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(7)
thf(fact_1125_rrb__simps_I8_J,axiom,
! [Y3: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y3 @ Qy ) )
= ( ( ( member_nat @ Y3 @ ( relational_fv_a_b @ Qy ) )
=> ? [X2: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y3 @ Qy @ X2 ) )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(8)
thf(fact_1126_qps__rrb,axiom,
! [Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q3 @ ( relational_qps_a_b @ G ) )
=> ( relational_rrb_a_b @ Q3 ) ) ).
% qps_rrb
thf(fact_1127_ex__cov,axiom,
! [Q3: relational_fmla_a_b,X4: nat] :
( ( relational_rrb_a_b @ Q3 )
=> ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_cov_a_b @ X4 @ Q3 @ X_1 ) ) ) ).
% ex_cov
thf(fact_1128_rrb__def,axiom,
( relational_rrb_a_b
= ( ^ [Q: relational_fmla_a_b] :
! [Y5: nat,Qy2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y5 @ Qy2 ) @ ( relational_sub_a_b @ Q ) )
=> ? [X2: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y5 @ Qy2 @ X2 ) ) ) ) ).
% rrb_def
thf(fact_1129_finite__if__eq__beyond__finite,axiom,
! [S: set_nat,S2: set_nat] :
( ( finite_finite_nat @ S )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [S3: set_nat] :
( ( minus_minus_set_nat @ S3 @ S )
= ( minus_minus_set_nat @ S2 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_1130_gen__finite,axiom,
! [X4: nat,Q3: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X4 @ Q3 @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% gen_finite
thf(fact_1131_finite__csts__term,axiom,
! [T2: relational_term_nat] : ( finite_finite_nat @ ( relati694035416245573993rm_nat @ T2 ) ) ).
% finite_csts_term
thf(fact_1132_infinite__surj,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_1133_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1134_finite__Collect__conjI,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
| ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
& ( Q3 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1135_finite__Collect__conjI,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q3 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q3 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1136_finite__Collect__disjI,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
| ( Q3 @ X ) ) ) )
= ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
& ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1137_finite__Collect__disjI,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q3 @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1138_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1139_finite__Collect__bounded__ex,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
? [Y5: relational_fmla_a_b] :
( ( P @ Y5 )
& ( Q3 @ X @ Y5 ) ) ) )
= ( ! [Y5: relational_fmla_a_b] :
( ( P @ Y5 )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( Q3 @ X @ Y5 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1140_finite__Collect__bounded__ex,axiom,
! [P: relational_fmla_a_b > $o,Q3: nat > relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y5: relational_fmla_a_b] :
( ( P @ Y5 )
& ( Q3 @ X @ Y5 ) ) ) )
= ( ! [Y5: relational_fmla_a_b] :
( ( P @ Y5 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q3 @ X @ Y5 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1141_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q3: relational_fmla_a_b > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
? [Y5: nat] :
( ( P @ Y5 )
& ( Q3 @ X @ Y5 ) ) ) )
= ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( Q3 @ X @ Y5 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1142_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q3: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y5: nat] :
( ( P @ Y5 )
& ( Q3 @ X @ Y5 ) ) ) )
= ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q3 @ X @ Y5 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1143_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1144_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_1145_not__finite__existsD,axiom,
! [P: relational_fmla_a_b > $o] :
( ~ ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ? [X_1: relational_fmla_a_b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1146_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1147_pigeonhole__infinite__rel,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_nat,R: relational_fmla_a_b > nat > $o] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A5 @ A )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1148_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1149_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1150_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1151_infinite__imp__nonempty,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_1152_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1153_finite_OemptyI,axiom,
finite5600759454172676150la_a_b @ bot_bo4495933725496725865la_a_b ).
% finite.emptyI
thf(fact_1154_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1155_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_1156_infinite__super,axiom,
! [S: set_nat,T6: set_nat] :
( ( ord_less_eq_set_nat @ S @ T6 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T6 ) ) ) ).
% infinite_super
thf(fact_1157_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_1158_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1159_pigeonhole__infinite,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ ( image_341122591648980342_b_nat @ F @ A ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A5 @ A )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1160_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1161_finite__image__set,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_1162_finite__image__set,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: relational_fmla_a_b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_1163_finite__image__set,axiom,
! [P: nat > $o,F: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_1164_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_1165_finite__image__set2,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1166_finite__image__set2,axiom,
! [P: relational_fmla_a_b > $o,Q3: relational_fmla_a_b > $o,F: relational_fmla_a_b > relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1167_finite__image__set2,axiom,
! [P: relational_fmla_a_b > $o,Q3: nat > $o,F: relational_fmla_a_b > nat > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q3 ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: relational_fmla_a_b,Y5: nat] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1168_finite__image__set2,axiom,
! [P: relational_fmla_a_b > $o,Q3: nat > $o,F: relational_fmla_a_b > nat > nat] :
( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: relational_fmla_a_b,Y5: nat] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1169_finite__image__set2,axiom,
! [P: nat > $o,Q3: relational_fmla_a_b > $o,F: nat > relational_fmla_a_b > relational_fmla_a_b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: nat,Y5: relational_fmla_a_b] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1170_finite__image__set2,axiom,
! [P: nat > $o,Q3: relational_fmla_a_b > $o,F: nat > relational_fmla_a_b > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y5: relational_fmla_a_b] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1171_finite__image__set2,axiom,
! [P: nat > $o,Q3: nat > $o,F: nat > nat > relational_fmla_a_b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q3 ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [Uu: relational_fmla_a_b] :
? [X: nat,Y5: nat] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1172_finite__image__set2,axiom,
! [P: nat > $o,Q3: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y5: nat] :
( ( Uu
= ( F @ X @ Y5 ) )
& ( P @ X )
& ( Q3 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1173_finite__set__of__finite__funs,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,D: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( finite831239660936554612la_a_b
@ ( collec5041345499257167282la_a_b
@ ^ [F2: relational_fmla_a_b > relational_fmla_a_b] :
! [X: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F2 @ X ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( F2 @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_1174_finite__set__of__finite__funs,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_nat,D: nat] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite3501642712278012965_b_nat
@ ( collec115744212963325795_b_nat
@ ^ [F2: relational_fmla_a_b > nat] :
! [X: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( F2 @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_1175_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_Re381260168593705685la_a_b,D: relational_fmla_a_b] :
( ( finite_finite_nat @ A )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( finite7579272753630036901la_a_b
@ ( collec4193374254315349731la_a_b
@ ^ [F2: nat > relational_fmla_a_b] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F2 @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F2 @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_1176_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat,D: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F2 @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_1177_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1178_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1179_infinite__finite__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A: set_Re381260168593705685la_a_b] :
( ! [A6: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1180_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1181_finite__ne__induct,axiom,
! [F4: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( F4 != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b] : ( P @ ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
=> ( ! [X3: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( F3 != bot_bo4495933725496725865la_a_b )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1182_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1183_finite__induct,axiom,
! [F4: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1184_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1185_finite_Osimps,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A5: set_Re381260168593705685la_a_b] :
( ( A5 = bot_bo4495933725496725865la_a_b )
| ? [A3: set_Re381260168593705685la_a_b,B4: relational_fmla_a_b] :
( ( A5
= ( insert7010464514620295119la_a_b @ B4 @ A3 ) )
& ( finite5600759454172676150la_a_b @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1186_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B4: nat] :
( ( A5
= ( insert_nat @ B4 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1187_finite_Ocases,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ~ ! [A6: set_Re381260168593705685la_a_b] :
( ? [A4: relational_fmla_a_b] :
( A2
= ( insert7010464514620295119la_a_b @ A4 @ A6 ) )
=> ~ ( finite5600759454172676150la_a_b @ A6 ) ) ) ) ).
% finite.cases
thf(fact_1188_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_1189_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1190_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1191_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A )
& ( finite_finite_nat @ C6 )
& ( B
= ( image_nat_nat @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1192_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1193_finite__subset__induct_H,axiom,
! [F4: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( ord_le4112832032246704949la_a_b @ F4 @ A )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A4: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( member4680049679412964150la_a_b @ A4 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert7010464514620295119la_a_b @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1194_finite__subset__induct_H,axiom,
! [F4: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1195_finite__subset__induct,axiom,
! [F4: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( ord_le4112832032246704949la_a_b @ F4 @ A )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A4: relational_fmla_a_b,F3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( member4680049679412964150la_a_b @ A4 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert7010464514620295119la_a_b @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1196_finite__subset__induct,axiom,
! [F4: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1197_finite__empty__induct,axiom,
! [A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( P @ A )
=> ( ! [A4: relational_fmla_a_b,A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( member4680049679412964150la_a_b @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ A4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ( P @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_1198_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1199_infinite__coinduct,axiom,
! [X5: set_Re381260168593705685la_a_b > $o,A: set_Re381260168593705685la_a_b] :
( ( X5 @ A )
=> ( ! [A6: set_Re381260168593705685la_a_b] :
( ( X5 @ A6 )
=> ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A6 )
& ( ( X5 @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) )
| ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ~ ( finite5600759454172676150la_a_b @ A ) ) ) ).
% infinite_coinduct
thf(fact_1200_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1201_infinite__remove,axiom,
! [S: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% infinite_remove
thf(fact_1202_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1203_finite__remove__induct,axiom,
! [B: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( A6 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A6 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1204_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1205_remove__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ( ~ ( finite5600759454172676150la_a_b @ B )
=> ( P @ B ) )
=> ( ! [A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( A6 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A6 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1206_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1207_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1208_finite__ranking__induct,axiom,
! [S: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,S4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S4 )
=> ( ! [Y6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1209_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1210_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q ) @ Q ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_1211_eval__cong,axiom,
! [Q3: relational_fmla_a_b,Q5: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( ( relational_fv_a_b @ Q3 )
= ( relational_fv_a_b @ Q5 ) )
=> ( ! [Sigma3: nat > a] :
( ( relational_sat_a_b @ Q3 @ I @ Sigma3 )
= ( relational_sat_a_b @ Q5 @ I @ Sigma3 ) )
=> ( ( relational_eval_a_b @ Q3 @ I )
= ( relational_eval_a_b @ Q5 @ I ) ) ) ) ).
% eval_cong
thf(fact_1212_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q3: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q3 ) @ ( relational_fv_a_b @ Q5 ) )
=> ( ( relational_equiv_a_b @ Q3 @ Q5 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q5 ) @ Q3 @ I )
= ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_1213_infinite__eval__Conj,axiom,
! [X4: nat,Q3: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Y3: nat] :
( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
=> ( ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q3 @ I ) )
=> ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) ) @ I ) ) ) ) ).
% infinite_eval_Conj
thf(fact_1214_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q3: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q3 )
= ( relational_fv_a_b @ Q5 ) )
=> ( ( relational_equiv_a_b @ Q3 @ Q5 )
=> ( ( relational_eval_a_b @ Q3 @ I )
= ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_1215_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1216_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M: nat] :
( ( P @ X4 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1217_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1218_diff__diff__cancel,axiom,
! [I2: nat,N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1219_eq__diff__iff,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N4 @ K ) )
= ( M4 = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_1220_le__diff__iff,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( ord_less_eq_nat @ M4 @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_1221_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( minus_minus_nat @ M4 @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1222_diff__le__mono,axiom,
! [M4: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L2 ) @ ( minus_minus_nat @ N4 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1223_diff__le__self,axiom,
! [M4: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N4 ) @ M4 ) ).
% diff_le_self
thf(fact_1224_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1225_diff__le__mono2,axiom,
! [M4: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N4 ) @ ( minus_minus_nat @ L2 @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1226_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_1227_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1228_eq__imp__le,axiom,
! [M4: nat,N4: nat] :
( ( M4 = N4 )
=> ( ord_less_eq_nat @ M4 @ N4 ) ) ).
% eq_imp_le
thf(fact_1229_le__antisym,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M4 )
=> ( M4 = N4 ) ) ) ).
% le_antisym
thf(fact_1230_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_1231_nat__le__linear,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
| ( ord_less_eq_nat @ N4 @ M4 ) ) ).
% nat_le_linear
thf(fact_1232_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_1233_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1234_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1235_cpropagated__simps_I7_J,axiom,
! [X4: nat,Q3: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q3 ) )
= ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ).
% cpropagated_simps(7)
thf(fact_1236_psubset__imp__ex__mem,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B )
=> ? [B3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ B3 @ ( minus_4077726661957047470la_a_b @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1237_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1238_leD,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y3 ) ) ).
% leD
thf(fact_1239_leI,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% leI
thf(fact_1240_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1241_antisym__conv1,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1242_antisym__conv2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1243_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1244_not__le__imp__less,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ord_less_nat @ X4 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_1245_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1246_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1247_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1248_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1249_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1250_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1251_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1252_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1253_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1254_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1255_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1256_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1257_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1258_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1259_linorder__not__le,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_not_le
thf(fact_1260_linorder__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_not_less
thf(fact_1261_order__less__imp__le,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1262_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1263_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1264_order__le__less__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1265_order__less__le__trans,axiom,
! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1266_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1267_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1268_linorder__le__less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_1269_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1270_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1271_cpropagated__sub,axiom,
! [Q3: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q3 )
=> ( ( member4680049679412964150la_a_b @ Q5 @ ( relational_sub_a_b @ Q3 ) )
=> ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).
% cpropagated_sub
thf(fact_1272_psubsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% psubsetD
thf(fact_1273_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_1274_cpropagated__flat__DisjD,axiom,
! [Q5: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q5 @ ( restri569617705344514291sj_a_b @ Q3 ) )
=> ( ( relati1591879772219623554ed_a_b @ Q3 )
=> ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).
% cpropagated_flat_DisjD
% Helper facts (13)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y3: nat] :
( ( if_nat @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y3: nat] :
( ( if_nat @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_M_Eo_J_T,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( if_nat_o @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_M_Eo_J_T,axiom,
! [X4: nat > $o,Y3: nat > $o] :
( ( if_nat_o @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X4: relational_fmla_a_b,Y3: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X4: relational_fmla_a_b,Y3: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J_T,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( if_Rel8262680441166333110_a_b_o @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J_T,axiom,
! [X4: relational_fmla_a_b > $o,Y3: relational_fmla_a_b > $o] :
( ( if_Rel8262680441166333110_a_b_o @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X4: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
( ( collec3419995626248312948la_a_b
@ ^ [Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q @ q2 )
& ( member_nat @ x
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ~ ? [X2: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X2 ) ) ) ) ) )
= bot_bo4495933725496725865la_a_b ) ).
thf(conj_1,hypothesis,
member_nat @ x @ ( relational_fv_a_b @ q ) ).
thf(conj_2,hypothesis,
member4680049679412964150la_a_b @ q @ q2 ).
thf(conj_3,conjecture,
? [G6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ x @ q @ G6 ) ).
%------------------------------------------------------------------------------