TPTP Problem File: SLH0668^1.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 : FO_Theory_Rewriting/0055_NF/prob_00087_003116__18414538_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1489 ( 409 unt; 223 typ; 0 def)
% Number of atoms : 3977 ( 984 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12333 ( 373 ~; 57 |; 415 &;9522 @)
% ( 0 <=>;1966 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 1097 (1097 >; 0 *; 0 +; 0 <<)
% Number of symbols : 202 ( 199 usr; 6 con; 0-3 aty)
% Number of variables : 3927 ( 405 ^;3305 !; 217 ?;3927 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:01:09.540
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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% Explicit typings (199)
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collec6157462517194527428term_a: ( set_fs1788988886788723183term_a > $o ) > set_se373288948044503333term_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
collec152922063390942756term_a: ( set_li5007820469839914319term_a > $o ) > set_se6278367998894210181term_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OPow_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
pow_fs3226181871949320924term_a: set_fs1788988886788723183term_a > set_se373288948044503333term_a ).
thf(sy_c_Set_OPow_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
pow_li3987379423859365436term_a: set_li5007820469839914319term_a > set_se6278367998894210181term_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_5390476545841808249term_a: ( fset_Bot_bot_term_a > fset_Bot_bot_term_a ) > set_fs1788988886788723183term_a > set_fs1788988886788723183term_a ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_6151674097751852761term_a: ( fset_Bot_bot_term_a > list_Bot_bot_term_a ) > set_fs1788988886788723183term_a > set_li5007820469839914319term_a ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__Nat__Onat,type,
image_8474690510679110544_a_nat: ( fset_Bot_bot_term_a > nat ) > set_fs1788988886788723183term_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_4385135741129320153term_a: ( list_Bot_bot_term_a > fset_Bot_bot_term_a ) > set_li5007820469839914319term_a > set_fs1788988886788723183term_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_5146333293039364665term_a: ( list_Bot_bot_term_a > list_Bot_bot_term_a ) > set_li5007820469839914319term_a > set_li5007820469839914319term_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__Nat__Onat,type,
image_1574777119431997168_a_nat: ( list_Bot_bot_term_a > nat ) > set_li5007820469839914319term_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_1785693973830655120term_a: ( nat > fset_Bot_bot_term_a ) > set_nat > set_fs1788988886788723183term_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
image_2546891525740699632term_a: ( nat > list_Bot_bot_term_a ) > set_nat > set_li5007820469839914319term_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Utils_Otrancl__listp_001t__Bot____Terms__Obot____term_Itf__a_J,type,
trancl4041116690992654595term_a: ( bot_bot_term_a > bot_bot_term_a > $o ) > list_Bot_bot_term_a > list_Bot_bot_term_a > $o ).
thf(sy_c_Utils_Otrancl__listp_001t__Nat__Onat,type,
trancl_listp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_member_001t__Bot____Terms__Obot____term_Itf__a_J,type,
member2723211829317350432term_a: bot_bot_term_a > set_Bot_bot_term_a > $o ).
thf(sy_c_member_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
member2089387167371741008term_a: fset_Bot_bot_term_a > set_fs1788988886788723183term_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
member2850584719281785520term_a: list_Bot_bot_term_a > set_li5007820469839914319term_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
member2313390693227633030term_a: set_fs1788988886788723183term_a > set_se373288948044503333term_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
member5532222276278824166term_a: set_li5007820469839914319term_a > set_se6278367998894210181term_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_R,type,
r: fset_term_a_b ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1262)
thf(fact_0_finite__Collect__conjI,axiom,
! [P: fset_Bot_bot_term_a > $o,Q: fset_Bot_bot_term_a > $o] :
( ( ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ P ) )
| ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ Q ) ) )
=> ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1_finite__Collect__conjI,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ( ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ P ) )
| ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ Q ) ) )
=> ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3_finite__Collect__disjI,axiom,
! [P: fset_Bot_bot_term_a > $o,Q: fset_Bot_bot_term_a > $o] :
( ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ P ) )
& ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_4_finite__Collect__disjI,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ P ) )
& ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_6_fsubset__antisym,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( ord_le7216997114146882585term_a @ B @ A )
=> ( A = B ) ) ) ).
% fsubset_antisym
thf(fact_7_order__refl,axiom,
! [X2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ X2 @ X2 ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_9_dual__order_Orefl,axiom,
! [A2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_11_finite__has__maximal2,axiom,
! [A: set_fs1788988886788723183term_a,A2: fset_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( member2089387167371741008term_a @ A2 @ A )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
& ( ord_le7216997114146882585term_a @ A2 @ X3 )
& ! [Xa: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Xa @ A )
=> ( ( ord_le7216997114146882585term_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_12_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_13_finite__has__minimal2,axiom,
! [A: set_fs1788988886788723183term_a,A2: fset_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( member2089387167371741008term_a @ A2 @ A )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
& ( ord_le7216997114146882585term_a @ X3 @ A2 )
& ! [Xa: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Xa @ A )
=> ( ( ord_le7216997114146882585term_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_14_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_15_not__finite__existsD,axiom,
! [P: fset_Bot_bot_term_a > $o] :
( ~ ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ P ) )
=> ? [X_1: fset_Bot_bot_term_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_16_not__finite__existsD,axiom,
! [P: list_Bot_bot_term_a > $o] :
( ~ ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ P ) )
=> ? [X_1: list_Bot_bot_term_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_17_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_18_pigeonhole__infinite__rel,axiom,
! [A: set_li5007820469839914319term_a,B: set_fs1788988886788723183term_a,R: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite8953276157157055568term_a @ B )
=> ( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ? [Xa: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ B )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_19_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_fs1788988886788723183term_a,R: nat > fset_Bot_bot_term_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite8953276157157055568term_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_20_pigeonhole__infinite__rel,axiom,
! [A: set_fs1788988886788723183term_a,B: set_li5007820469839914319term_a,R: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite491101672212324272term_a @ B )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ? [Xa: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ B )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_21_pigeonhole__infinite__rel,axiom,
! [A: set_fs1788988886788723183term_a,B: set_nat,R: fset_Bot_bot_term_a > nat > $o] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_22_pigeonhole__infinite__rel,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a,R: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite8953276157157055568term_a @ B )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ? [Xa: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ B )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_23_pigeonhole__infinite__rel,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a,R: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite491101672212324272term_a @ B )
=> ( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ? [Xa: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ B )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_24_pigeonhole__infinite__rel,axiom,
! [A: set_li5007820469839914319term_a,B: set_nat,R: list_Bot_bot_term_a > nat > $o] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_25_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_li5007820469839914319term_a,R: nat > list_Bot_bot_term_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite491101672212324272term_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_26_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
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_27_exists__fset__of__list,axiom,
! [S: fset_Bot_bot_term_a] :
? [Xs: list_Bot_bot_term_a] :
( ( fset_o7715858782369100227term_a @ Xs )
= S ) ).
% exists_fset_of_list
thf(fact_28_fequalityE,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( A = B )
=> ~ ( ( ord_le7216997114146882585term_a @ A @ B )
=> ~ ( ord_le7216997114146882585term_a @ B @ A ) ) ) ).
% fequalityE
thf(fact_29_finite__Collect__subsets,axiom,
! [A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ( finite1423361352422162918term_a
@ ( collec152922063390942756term_a
@ ^ [B2: set_li5007820469839914319term_a] : ( ord_le549404264473380271term_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_30_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_31_finite__Collect__subsets,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( finite7427901806225747590term_a
@ ( collec6157462517194527428term_a
@ ^ [B2: set_fs1788988886788723183term_a] : ( ord_le6553944718276964943term_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_32_rev__finite__subset,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ A @ B )
=> ( finite491101672212324272term_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_33_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_34_rev__finite__subset,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ A @ B )
=> ( finite8953276157157055568term_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_35_infinite__super,axiom,
! [S: set_li5007820469839914319term_a,T: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ S @ T )
=> ( ~ ( finite491101672212324272term_a @ S )
=> ~ ( finite491101672212324272term_a @ T ) ) ) ).
% infinite_super
thf(fact_36_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_37_infinite__super,axiom,
! [S: set_fs1788988886788723183term_a,T: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ S @ T )
=> ( ~ ( finite8953276157157055568term_a @ S )
=> ~ ( finite8953276157157055568term_a @ T ) ) ) ).
% infinite_super
thf(fact_38_finite__subset,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( finite491101672212324272term_a @ B )
=> ( finite491101672212324272term_a @ A ) ) ) ).
% finite_subset
thf(fact_39_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_40_finite__subset,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( finite8953276157157055568term_a @ B )
=> ( finite8953276157157055568term_a @ A ) ) ) ).
% finite_subset
thf(fact_41_order__antisym__conv,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Y @ X2 )
=> ( ( ord_le7216997114146882585term_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_42_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_43_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_44_ord__le__eq__subst,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__le__eq__subst,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > nat,C: nat] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_48_ord__eq__le__subst,axiom,
! [A2: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A2: nat,F: fset_Bot_bot_term_a > nat,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A2: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_53_order__eq__refl,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( X2 = Y )
=> ( ord_le7216997114146882585term_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_54_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_55_order__subst2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ ( F @ B3 ) @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_56_order__subst2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > nat,C: nat] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_57_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_58_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_59_order__subst1,axiom,
! [A2: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_60_order__subst1,axiom,
! [A2: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,B3: nat,C: nat] :
( ( ord_le7216997114146882585term_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7216997114146882585term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_61_order__subst1,axiom,
! [A2: nat,F: fset_Bot_bot_term_a > nat,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_62_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_63_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_Bot_bot_term_a,Z: fset_Bot_bot_term_a] : ( Y3 = Z ) )
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A3 @ B4 )
& ( ord_le7216997114146882585term_a @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_64_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_65_antisym,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_66_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_67_dual__order_Otrans,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( ( ord_le7216997114146882585term_a @ C @ B3 )
=> ( ord_le7216997114146882585term_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_68_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_69_dual__order_Oantisym,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_70_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_71_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: fset_Bot_bot_term_a,Z: fset_Bot_bot_term_a] : ( Y3 = Z ) )
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B4 @ A3 )
& ( ord_le7216997114146882585term_a @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_72_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_73_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_74_order__trans,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a,Z2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( ord_le7216997114146882585term_a @ Y @ Z2 )
=> ( ord_le7216997114146882585term_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_75_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_76_order_Otrans,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ord_le7216997114146882585term_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_77_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_78_order__antisym,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( ord_le7216997114146882585term_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_79_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_80_ord__le__eq__trans,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le7216997114146882585term_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_81_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_82_ord__eq__le__trans,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( A2 = B3 )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ord_le7216997114146882585term_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_83_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_84_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_Bot_bot_term_a,Z: fset_Bot_bot_term_a] : ( Y3 = Z ) )
= ( ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X @ Y4 )
& ( ord_le7216997114146882585term_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_85_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_86_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_87_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_88_fset__eq__fsubset,axiom,
( ( ^ [Y3: fset_Bot_bot_term_a,Z: fset_Bot_bot_term_a] : ( Y3 = Z ) )
= ( ^ [A5: fset_Bot_bot_term_a,B2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A5 @ B2 )
& ( ord_le7216997114146882585term_a @ B2 @ A5 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_89_mem__Collect__eq,axiom,
! [A2: fset_Bot_bot_term_a,P: fset_Bot_bot_term_a > $o] :
( ( member2089387167371741008term_a @ A2 @ ( collec3259196342482385038term_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A2: list_Bot_bot_term_a,P: list_Bot_bot_term_a > $o] :
( ( member2850584719281785520term_a @ A2 @ ( collec4020393894392429550term_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A: set_li5007820469839914319term_a] :
( ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_95_Collect__cong,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ! [X3: list_Bot_bot_term_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec4020393894392429550term_a @ P )
= ( collec4020393894392429550term_a @ Q ) ) ) ).
% Collect_cong
thf(fact_96_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_97_fsubset__trans,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( ord_le7216997114146882585term_a @ B @ C2 )
=> ( ord_le7216997114146882585term_a @ A @ C2 ) ) ) ).
% fsubset_trans
thf(fact_98_fsubset__refl,axiom,
! [A: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ A @ A ) ).
% fsubset_refl
thf(fact_99_fequalityD2,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( A = B )
=> ( ord_le7216997114146882585term_a @ B @ A ) ) ).
% fequalityD2
thf(fact_100_fequalityD1,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( A = B )
=> ( ord_le7216997114146882585term_a @ A @ B ) ) ).
% fequalityD1
thf(fact_101_Greatest__equality,axiom,
! [P: fset_Bot_bot_term_a > $o,X2: fset_Bot_bot_term_a] :
( ( P @ X2 )
=> ( ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ Y2 @ X2 ) )
=> ( ( order_8550835685197085856term_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_102_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_103_GreatestI2__order,axiom,
! [P: fset_Bot_bot_term_a > $o,X2: fset_Bot_bot_term_a,Q: fset_Bot_bot_term_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ Y2 @ X2 ) )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( P @ X3 )
=> ( ! [Y5: fset_Bot_bot_term_a] :
( ( P @ Y5 )
=> ( ord_le7216997114146882585term_a @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_8550835685197085856term_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_104_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_105_neq__if__length__neq,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs2 )
!= ( size_s1103687553077312429term_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_106_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_107_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_Bot_bot_term_a] :
( ( size_s1103687553077312429term_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_108_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_109_size__neq__size__imp__neq,axiom,
! [X2: list_Bot_bot_term_a,Y: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ X2 )
!= ( size_s1103687553077312429term_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_110_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_111_finite__Collect__less__eq,axiom,
! [Q: list_Bot_bot_term_a > $o,P: list_Bot_bot_term_a > $o] :
( ( ord_le8931594491441365398rm_a_o @ Q @ P )
=> ( ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ P ) )
=> ( finite491101672212324272term_a @ ( collec4020393894392429550term_a @ Q ) ) ) ) ).
% finite_Collect_less_eq
thf(fact_112_finite__Collect__less__eq,axiom,
! [Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_nat_o @ Q @ P )
=> ( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_less_eq
thf(fact_113_finite__Collect__less__eq,axiom,
! [Q: fset_Bot_bot_term_a > $o,P: fset_Bot_bot_term_a > $o] :
( ( ord_le6418193777846492406rm_a_o @ Q @ P )
=> ( ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ P ) )
=> ( finite8953276157157055568term_a @ ( collec3259196342482385038term_a @ Q ) ) ) ) ).
% finite_Collect_less_eq
thf(fact_114_infinite__imp__elem,axiom,
! [A: set_li5007820469839914319term_a] :
( ~ ( finite491101672212324272term_a @ A )
=> ? [X3: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X3 @ A ) ) ).
% infinite_imp_elem
thf(fact_115_infinite__imp__elem,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ? [X3: nat] : ( member_nat @ X3 @ A ) ) ).
% infinite_imp_elem
thf(fact_116_infinite__imp__elem,axiom,
! [A: set_fs1788988886788723183term_a] :
( ~ ( finite8953276157157055568term_a @ A )
=> ? [X3: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X3 @ A ) ) ).
% infinite_imp_elem
thf(fact_117_le__rel__bool__arg__iff,axiom,
( ord_le1167812134399094754term_a
= ( ^ [X4: $o > fset_Bot_bot_term_a,Y6: $o > fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le7216997114146882585term_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_118_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_119_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_120_verit__comp__simplify1_I2_J,axiom,
! [A2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_121_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_122_fset__of__list__subset,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs2 ) @ ( set_Bot_bot_term_a2 @ Ys ) )
=> ( ord_le7216997114146882585term_a @ ( fset_o7715858782369100227term_a @ Xs2 ) @ ( fset_o7715858782369100227term_a @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_123_Fpow__def,axiom,
( finite2823898951475608883term_a
= ( ^ [A5: set_li5007820469839914319term_a] :
( collec152922063390942756term_a
@ ^ [X4: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ X4 @ A5 )
& ( finite491101672212324272term_a @ X4 ) ) ) ) ) ).
% Fpow_def
thf(fact_124_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A5: set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ A5 )
& ( finite_finite_nat @ X4 ) ) ) ) ) ).
% Fpow_def
thf(fact_125_Fpow__def,axiom,
( finite2062701399565564371term_a
= ( ^ [A5: set_fs1788988886788723183term_a] :
( collec6157462517194527428term_a
@ ^ [X4: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ X4 @ A5 )
& ( finite8953276157157055568term_a @ X4 ) ) ) ) ) ).
% Fpow_def
thf(fact_126_List_Ofinite__set,axiom,
! [Xs2: list_l7107345091518559913term_a] : ( finite491101672212324272term_a @ ( set_li5239659345660059972term_a @ Xs2 ) ) ).
% List.finite_set
thf(fact_127_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_128_List_Ofinite__set,axiom,
! [Xs2: list_f3888513508467368777term_a] : ( finite8953276157157055568term_a @ ( set_fs4478461793750015460term_a @ Xs2 ) ) ).
% List.finite_set
thf(fact_129_subset__code_I1_J,axiom,
! [Xs2: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_130_subset__code_I1_J,axiom,
! [Xs2: list_f3888513508467368777term_a,B: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs2 ) @ B )
= ( ! [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ( member2089387167371741008term_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_131_subset__code_I1_J,axiom,
! [Xs2: list_l7107345091518559913term_a,B: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs2 ) @ B )
= ( ! [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ( member2850584719281785520term_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_132_finite__list,axiom,
! [A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ? [Xs: list_l7107345091518559913term_a] :
( ( set_li5239659345660059972term_a @ Xs )
= A ) ) ).
% finite_list
thf(fact_133_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs: list_nat] :
( ( set_nat2 @ Xs )
= A ) ) ).
% finite_list
thf(fact_134_finite__list,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ? [Xs: list_f3888513508467368777term_a] :
( ( set_fs4478461793750015460term_a @ Xs )
= A ) ) ).
% finite_list
thf(fact_135_finite__lists__length__eq,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ( finite491101672212324272term_a @ A )
=> ( finite366370572512868288term_a
@ ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs3 ) @ A )
& ( ( size_s1109596070125842749term_a @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_136_finite__lists__length__eq,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ( finite8953276157157055568term_a @ A )
=> ( finite6370911026316452960term_a
@ ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs3 ) @ A )
& ( ( size_s7114136523929427421term_a @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_137_finite__lists__length__eq,axiom,
! [A: set_Bot_bot_term_a,N: nat] :
( ( finite4019351022360529184term_a @ A )
=> ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs3 ) @ A )
& ( ( size_s1103687553077312429term_a @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_138_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_139_finite__lists__length__le,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ( finite491101672212324272term_a @ A )
=> ( finite366370572512868288term_a
@ ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s1109596070125842749term_a @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_140_finite__lists__length__le,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ( finite8953276157157055568term_a @ A )
=> ( finite6370911026316452960term_a
@ ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s7114136523929427421term_a @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_141_finite__lists__length__le,axiom,
! [A: set_Bot_bot_term_a,N: nat] :
( ( finite4019351022360529184term_a @ A )
=> ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_142_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_143_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_144_less__eq__set__def,axiom,
( ord_le6553944718276964943term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( ord_le6418193777846492406rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A5 )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_145_less__eq__set__def,axiom,
( ord_le549404264473380271term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( ord_le8931594491441365398rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A5 )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_146_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_147_pred__subset__eq,axiom,
! [R: set_fs1788988886788723183term_a,S: set_fs1788988886788723183term_a] :
( ( ord_le6418193777846492406rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ R )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ S ) )
= ( ord_le6553944718276964943term_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_148_pred__subset__eq,axiom,
! [R: set_li5007820469839914319term_a,S: set_li5007820469839914319term_a] :
( ( ord_le8931594491441365398rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ R )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ S ) )
= ( ord_le549404264473380271term_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_149_set__generate__lists,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( set_li5239659345660059972term_a @ ( missin698130009775865202term_a @ N @ Xs2 ) )
= ( collec4020393894392429550term_a
@ ^ [As: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ As )
= N )
& ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ As ) @ ( set_Bot_bot_term_a2 @ Xs2 ) ) ) ) ) ).
% set_generate_lists
thf(fact_150_set__generate__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( missin2047014633743487673ts_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [As: list_nat] :
( ( ( size_size_list_nat @ As )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ As ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_generate_lists
thf(fact_151_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_152_subsetI,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ( member2089387167371741008term_a @ X3 @ B ) )
=> ( ord_le6553944718276964943term_a @ A @ B ) ) ).
% subsetI
thf(fact_153_subsetI,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ( member2850584719281785520term_a @ X3 @ B ) )
=> ( ord_le549404264473380271term_a @ A @ B ) ) ).
% subsetI
thf(fact_154_finite__lists__distinct__length__eq,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ( finite491101672212324272term_a @ A )
=> ( finite366370572512868288term_a
@ ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( ( ( size_s1109596070125842749term_a @ Xs3 )
= N )
& ( distin8315165086625722720term_a @ Xs3 )
& ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_155_finite__lists__distinct__length__eq,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ( finite8953276157157055568term_a @ A )
=> ( finite6370911026316452960term_a
@ ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( ( ( size_s7114136523929427421term_a @ Xs3 )
= N )
& ( distin7553967534715678208term_a @ Xs3 )
& ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_156_finite__lists__distinct__length__eq,axiom,
! [A: set_Bot_bot_term_a,N: nat] :
( ( finite4019351022360529184term_a @ A )
=> ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs3 )
= N )
& ( distin8359246687647904464term_a @ Xs3 )
& ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_157_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ( distinct_nat @ Xs3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_158_set__n__lists,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( set_li5239659345660059972term_a @ ( n_list1546117208000512123term_a @ N @ Xs2 ) )
= ( collec4020393894392429550term_a
@ ^ [Ys2: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Ys2 )
= N )
& ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Ys2 ) @ ( set_Bot_bot_term_a2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_159_set__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_160_Collect__subset,axiom,
! [A: set_fs1788988886788723183term_a,P: fset_Bot_bot_term_a > $o] :
( ord_le6553944718276964943term_a
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_161_Collect__subset,axiom,
! [A: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o] :
( ord_le549404264473380271term_a
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_162_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_163_conj__subset__def,axiom,
! [A: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ( ord_le549404264473380271term_a @ A
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le549404264473380271term_a @ A @ ( collec4020393894392429550term_a @ P ) )
& ( ord_le549404264473380271term_a @ A @ ( collec4020393894392429550term_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_164_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_165_prop__restrict,axiom,
! [X2: fset_Bot_bot_term_a,Z3: set_fs1788988886788723183term_a,X5: set_fs1788988886788723183term_a,P: fset_Bot_bot_term_a > $o] :
( ( member2089387167371741008term_a @ X2 @ Z3 )
=> ( ( ord_le6553944718276964943term_a @ Z3
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_166_prop__restrict,axiom,
! [X2: list_Bot_bot_term_a,Z3: set_li5007820469839914319term_a,X5: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o] :
( ( member2850584719281785520term_a @ X2 @ Z3 )
=> ( ( ord_le549404264473380271term_a @ Z3
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_167_prop__restrict,axiom,
! [X2: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_168_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_169_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_170_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_171_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_172_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_173_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_174_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_175_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_176_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_177_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_178_finite__distinct,axiom,
! [X5: set_Bot_bot_term_a] :
( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( distin8359246687647904464term_a @ Xs3 )
& ( ( set_Bot_bot_term_a2 @ Xs3 )
= X5 ) ) ) ) ).
% finite_distinct
thf(fact_179_finite__distinct__list,axiom,
! [A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ? [Xs: list_l7107345091518559913term_a] :
( ( ( set_li5239659345660059972term_a @ Xs )
= A )
& ( distin8315165086625722720term_a @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_180_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= A )
& ( distinct_nat @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_181_finite__distinct__list,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ? [Xs: list_f3888513508467368777term_a] :
( ( ( set_fs4478461793750015460term_a @ Xs )
= A )
& ( distin7553967534715678208term_a @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_182_length__n__lists__elem,axiom,
! [Ys: list_Bot_bot_term_a,N: nat,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Ys @ ( set_li5239659345660059972term_a @ ( n_list1546117208000512123term_a @ N @ Xs2 ) ) )
=> ( ( size_s1103687553077312429term_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_183_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_184_finite__distinct__subset,axiom,
! [X5: set_Bot_bot_term_a] :
( ( finite4019351022360529184term_a @ X5 )
=> ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( distin8359246687647904464term_a @ Xs3 )
& ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs3 ) @ X5 ) ) ) ) ) ).
% finite_distinct_subset
thf(fact_185_finite__distinct__subset,axiom,
! [X5: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ X5 )
=> ( finite366370572512868288term_a
@ ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( ( distin8315165086625722720term_a @ Xs3 )
& ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs3 ) @ X5 ) ) ) ) ) ).
% finite_distinct_subset
thf(fact_186_finite__distinct__subset,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( distinct_nat @ Xs3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ X5 ) ) ) ) ) ).
% finite_distinct_subset
thf(fact_187_finite__distinct__subset,axiom,
! [X5: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ X5 )
=> ( finite6370911026316452960term_a
@ ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( ( distin7553967534715678208term_a @ Xs3 )
& ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs3 ) @ X5 ) ) ) ) ) ).
% finite_distinct_subset
thf(fact_188_Collect__mono__iff,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ( ord_le549404264473380271term_a @ ( collec4020393894392429550term_a @ P ) @ ( collec4020393894392429550term_a @ Q ) )
= ( ! [X: list_Bot_bot_term_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_189_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_190_Collect__mono,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ! [X3: list_Bot_bot_term_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le549404264473380271term_a @ ( collec4020393894392429550term_a @ P ) @ ( collec4020393894392429550term_a @ Q ) ) ) ).
% Collect_mono
thf(fact_191_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_192_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__iff,axiom,
( ord_le6553944718276964943term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
! [T2: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ T2 @ A5 )
=> ( member2089387167371741008term_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__iff,axiom,
( ord_le549404264473380271term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
! [T2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ T2 @ A5 )
=> ( member2850584719281785520term_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_195_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_196_subset__eq,axiom,
( ord_le6553944718276964943term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
! [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A5 )
=> ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_197_subset__eq,axiom,
( ord_le549404264473380271term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
! [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A5 )
=> ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_198_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_199_subsetD,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a,C: fset_Bot_bot_term_a] :
( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( member2089387167371741008term_a @ C @ A )
=> ( member2089387167371741008term_a @ C @ B ) ) ) ).
% subsetD
thf(fact_200_subsetD,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a,C: list_Bot_bot_term_a] :
( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( member2850584719281785520term_a @ C @ A )
=> ( member2850584719281785520term_a @ C @ B ) ) ) ).
% subsetD
thf(fact_201_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_202_in__mono,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a,X2: fset_Bot_bot_term_a] :
( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( member2089387167371741008term_a @ X2 @ A )
=> ( member2089387167371741008term_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a,X2: list_Bot_bot_term_a] :
( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( member2850584719281785520term_a @ X2 @ A )
=> ( member2850584719281785520term_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_204_infinite__imp__many__elems,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ~ ( finite491101672212324272term_a @ A )
=> ? [Xs: list_l7107345091518559913term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs ) @ A )
& ( ( size_s1109596070125842749term_a @ Xs )
= N )
& ( distin8315165086625722720term_a @ Xs ) ) ) ).
% infinite_imp_many_elems
thf(fact_205_infinite__imp__many__elems,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ~ ( finite8953276157157055568term_a @ A )
=> ? [Xs: list_f3888513508467368777term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs ) @ A )
& ( ( size_s7114136523929427421term_a @ Xs )
= N )
& ( distin7553967534715678208term_a @ Xs ) ) ) ).
% infinite_imp_many_elems
thf(fact_206_infinite__imp__many__elems,axiom,
! [A: set_Bot_bot_term_a,N: nat] :
( ~ ( finite4019351022360529184term_a @ A )
=> ? [Xs: list_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs ) @ A )
& ( ( size_s1103687553077312429term_a @ Xs )
= N )
& ( distin8359246687647904464term_a @ Xs ) ) ) ).
% infinite_imp_many_elems
thf(fact_207_infinite__imp__many__elems,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ( size_size_list_nat @ Xs )
= N )
& ( distinct_nat @ Xs ) ) ) ).
% infinite_imp_many_elems
thf(fact_208_Collect__restrict,axiom,
! [X5: set_fs1788988886788723183term_a,P: fset_Bot_bot_term_a > $o] :
( ord_le6553944718276964943term_a
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_209_Collect__restrict,axiom,
! [X5: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o] :
( ord_le549404264473380271term_a
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_210_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_211_subseqs__length__simple,axiom,
! [B3: list_Bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ B3 @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) )
=> ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ B3 ) @ ( size_s1103687553077312429term_a @ Xs2 ) ) ) ).
% subseqs_length_simple
thf(fact_212_subseqs__length__simple,axiom,
! [B3: list_nat,Xs2: list_nat] :
( ( member_list_nat @ B3 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ B3 ) @ ( size_size_list_nat @ Xs2 ) ) ) ).
% subseqs_length_simple
thf(fact_213_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_214_subset__Collect__iff,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a,P: fset_Bot_bot_term_a > $o] :
( ( ord_le6553944718276964943term_a @ B @ A )
=> ( ( ord_le6553944718276964943term_a @ B
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_215_subset__Collect__iff,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o] :
( ( ord_le549404264473380271term_a @ B @ A )
=> ( ( ord_le549404264473380271term_a @ B
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_216_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_217_subset__CollectI,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a,Q: fset_Bot_bot_term_a > $o,P: fset_Bot_bot_term_a > $o] :
( ( ord_le6553944718276964943term_a @ B @ A )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6553944718276964943term_a
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ B )
& ( Q @ X ) ) )
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_218_subset__CollectI,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a,Q: list_Bot_bot_term_a > $o,P: list_Bot_bot_term_a > $o] :
( ( ord_le549404264473380271term_a @ B @ A )
=> ( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le549404264473380271term_a
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ B )
& ( Q @ X ) ) )
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_219_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_220_in__set__product__lists__length,axiom,
! [Xs2: list_Bot_bot_term_a,Xss: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ Xs2 @ ( set_li5239659345660059972term_a @ ( produc196741067191133402term_a @ Xss ) ) )
=> ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_s1109596070125842749term_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_221_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_222_subseqs__refl,axiom,
! [Xs2: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ Xs2 @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_223_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_224_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_225_subseqs__distinctD,axiom,
! [Ys: list_Bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Ys @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) )
=> ( ( distin8359246687647904464term_a @ Xs2 )
=> ( distin8359246687647904464term_a @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_226_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_227_subseqs__length__simple__False,axiom,
! [B3: list_Bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ B3 @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) )
=> ~ ( ord_less_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ ( size_s1103687553077312429term_a @ B3 ) ) ) ).
% subseqs_length_simple_False
thf(fact_228_subseqs__length__simple__False,axiom,
! [B3: list_nat,Xs2: list_nat] :
( ( member_list_nat @ B3 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ~ ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ B3 ) ) ) ).
% subseqs_length_simple_False
thf(fact_229_product__lists__set,axiom,
! [Xss: list_list_nat] :
( ( set_list_nat2 @ ( product_lists_nat @ Xss ) )
= ( collect_list_nat
@ ^ [Xs3: list_nat] :
( list_a5155878676884504761st_nat
@ ^ [X: nat,Ys2: list_nat] : ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_230_product__lists__set,axiom,
! [Xss: list_l5937628333601154137term_a] :
( ( set_li9028854848070421492term_a @ ( produc7155444930543194890term_a @ Xss ) )
= ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( list_a4042977299686361739term_a
@ ^ [X: fset_Bot_bot_term_a,Ys2: list_f3888513508467368777term_a] : ( member2089387167371741008term_a @ X @ ( set_fs4478461793750015460term_a @ Ys2 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_231_product__lists__set,axiom,
! [Xss: list_l2619335347596085177term_a] :
( ( set_li3024314394266836820term_a @ ( produc7916642482453239402term_a @ Xss ) )
= ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( list_a4850894400288985419term_a
@ ^ [X: list_Bot_bot_term_a,Ys2: list_l7107345091518559913term_a] : ( member2850584719281785520term_a @ X @ ( set_li5239659345660059972term_a @ Ys2 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_232_product__lists__set,axiom,
! [Xss: list_l7107345091518559913term_a] :
( ( set_li5239659345660059972term_a @ ( produc196741067191133402term_a @ Xss ) )
= ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( list_a7305999888057299115term_a
@ ^ [X: bot_bot_term_a,Ys2: list_Bot_bot_term_a] : ( member2723211829317350432term_a @ X @ ( set_Bot_bot_term_a2 @ Ys2 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_233_length__n__lists,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( size_s1109596070125842749term_a @ ( n_list1546117208000512123term_a @ N @ Xs2 ) )
= ( power_power_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_234_length__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N @ Xs2 ) )
= ( power_power_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_235_finite__sorted__distinct__unique,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [X3: list_nat] :
( ( ( set_nat2 @ X3 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
& ( distinct_nat @ X3 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_236_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_237_subset__code_I2_J,axiom,
! [A: set_fs1788988886788723183term_a,Ys: list_f3888513508467368777term_a] :
( ( ord_le6553944718276964943term_a @ A @ ( coset_852189524471932294term_a @ Ys ) )
= ( ! [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ ( set_fs4478461793750015460term_a @ Ys ) )
=> ~ ( member2089387167371741008term_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_238_subset__code_I2_J,axiom,
! [A: set_li5007820469839914319term_a,Ys: list_l7107345091518559913term_a] :
( ( ord_le549404264473380271term_a @ A @ ( coset_1613387076381976806term_a @ Ys ) )
= ( ! [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ ( set_li5239659345660059972term_a @ Ys ) )
=> ~ ( member2850584719281785520term_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_239_image__eqI,axiom,
! [B3: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_240_image__eqI,axiom,
! [B3: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member2089387167371741008term_a @ B3 @ ( image_1785693973830655120term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_241_image__eqI,axiom,
! [B3: list_Bot_bot_term_a,F: nat > list_Bot_bot_term_a,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member2850584719281785520term_a @ B3 @ ( image_2546891525740699632term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_242_image__eqI,axiom,
! [B3: nat,F: fset_Bot_bot_term_a > nat,X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2089387167371741008term_a @ X2 @ A )
=> ( member_nat @ B3 @ ( image_8474690510679110544_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_243_image__eqI,axiom,
! [B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2089387167371741008term_a @ X2 @ A )
=> ( member2089387167371741008term_a @ B3 @ ( image_5390476545841808249term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_244_image__eqI,axiom,
! [B3: list_Bot_bot_term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a,X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2089387167371741008term_a @ X2 @ A )
=> ( member2850584719281785520term_a @ B3 @ ( image_6151674097751852761term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_245_image__eqI,axiom,
! [B3: nat,F: list_Bot_bot_term_a > nat,X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2850584719281785520term_a @ X2 @ A )
=> ( member_nat @ B3 @ ( image_1574777119431997168_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_246_image__eqI,axiom,
! [B3: fset_Bot_bot_term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a,X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2850584719281785520term_a @ X2 @ A )
=> ( member2089387167371741008term_a @ B3 @ ( image_4385135741129320153term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_247_image__eqI,axiom,
! [B3: list_Bot_bot_term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a,X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member2850584719281785520term_a @ X2 @ A )
=> ( member2850584719281785520term_a @ B3 @ ( image_5146333293039364665term_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_248_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_249_Un__iff,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( sup_su6676342596814492443term_a @ A @ B ) )
= ( ( member2089387167371741008term_a @ C @ A )
| ( member2089387167371741008term_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_250_Un__iff,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( sup_su671802143010907771term_a @ A @ B ) )
= ( ( member2850584719281785520term_a @ C @ A )
| ( member2850584719281785520term_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_251_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_252_UnCI,axiom,
! [C: fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( ~ ( member2089387167371741008term_a @ C @ B )
=> ( member2089387167371741008term_a @ C @ A ) )
=> ( member2089387167371741008term_a @ C @ ( sup_su6676342596814492443term_a @ A @ B ) ) ) ).
% UnCI
thf(fact_253_UnCI,axiom,
! [C: list_Bot_bot_term_a,B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( ~ ( member2850584719281785520term_a @ C @ B )
=> ( member2850584719281785520term_a @ C @ A ) )
=> ( member2850584719281785520term_a @ C @ ( sup_su671802143010907771term_a @ A @ B ) ) ) ).
% UnCI
thf(fact_254_image__ident,axiom,
! [Y7: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_255_finite__imageI,axiom,
! [F2: set_li5007820469839914319term_a,H: list_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( finite491101672212324272term_a @ F2 )
=> ( finite491101672212324272term_a @ ( image_5146333293039364665term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_256_finite__imageI,axiom,
! [F2: set_li5007820469839914319term_a,H: list_Bot_bot_term_a > nat] :
( ( finite491101672212324272term_a @ F2 )
=> ( finite_finite_nat @ ( image_1574777119431997168_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_257_finite__imageI,axiom,
! [F2: set_li5007820469839914319term_a,H: list_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( finite491101672212324272term_a @ F2 )
=> ( finite8953276157157055568term_a @ ( image_4385135741129320153term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_258_finite__imageI,axiom,
! [F2: set_nat,H: nat > list_Bot_bot_term_a] :
( ( finite_finite_nat @ F2 )
=> ( finite491101672212324272term_a @ ( image_2546891525740699632term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_259_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_260_finite__imageI,axiom,
! [F2: set_nat,H: nat > fset_Bot_bot_term_a] :
( ( finite_finite_nat @ F2 )
=> ( finite8953276157157055568term_a @ ( image_1785693973830655120term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_261_finite__imageI,axiom,
! [F2: set_fs1788988886788723183term_a,H: fset_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ F2 )
=> ( finite491101672212324272term_a @ ( image_6151674097751852761term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_262_finite__imageI,axiom,
! [F2: set_fs1788988886788723183term_a,H: fset_Bot_bot_term_a > nat] :
( ( finite8953276157157055568term_a @ F2 )
=> ( finite_finite_nat @ ( image_8474690510679110544_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_263_finite__imageI,axiom,
! [F2: set_fs1788988886788723183term_a,H: fset_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ F2 )
=> ( finite8953276157157055568term_a @ ( image_5390476545841808249term_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_264_finite__Un,axiom,
! [F2: set_li5007820469839914319term_a,G: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ ( sup_su671802143010907771term_a @ F2 @ G ) )
= ( ( finite491101672212324272term_a @ F2 )
& ( finite491101672212324272term_a @ G ) ) ) ).
% finite_Un
thf(fact_265_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_266_finite__Un,axiom,
! [F2: set_fs1788988886788723183term_a,G: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ ( sup_su6676342596814492443term_a @ F2 @ G ) )
= ( ( finite8953276157157055568term_a @ F2 )
& ( finite8953276157157055568term_a @ G ) ) ) ).
% finite_Un
thf(fact_267_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_268_Compr__image__eq,axiom,
! [F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: fset_Bot_bot_term_a > $o] :
( ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ ( image_5390476545841808249term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_5390476545841808249term_a @ F
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_269_Compr__image__eq,axiom,
! [F: list_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: fset_Bot_bot_term_a > $o] :
( ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ ( image_4385135741129320153term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_4385135741129320153term_a @ F
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_270_Compr__image__eq,axiom,
! [F: nat > fset_Bot_bot_term_a,A: set_nat,P: fset_Bot_bot_term_a > $o] :
( ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ ( image_1785693973830655120term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_1785693973830655120term_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_271_Compr__image__eq,axiom,
! [F: fset_Bot_bot_term_a > list_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: list_Bot_bot_term_a > $o] :
( ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ ( image_6151674097751852761term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_6151674097751852761term_a @ F
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_272_Compr__image__eq,axiom,
! [F: list_Bot_bot_term_a > list_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: list_Bot_bot_term_a > $o] :
( ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ ( image_5146333293039364665term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_5146333293039364665term_a @ F
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_273_Compr__image__eq,axiom,
! [F: nat > list_Bot_bot_term_a,A: set_nat,P: list_Bot_bot_term_a > $o] :
( ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ ( image_2546891525740699632term_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_2546891525740699632term_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_274_Compr__image__eq,axiom,
! [F: fset_Bot_bot_term_a > nat,A: set_fs1788988886788723183term_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_8474690510679110544_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_8474690510679110544_a_nat @ F
@ ( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_275_Compr__image__eq,axiom,
! [F: list_Bot_bot_term_a > nat,A: set_li5007820469839914319term_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_1574777119431997168_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_1574777119431997168_a_nat @ F
@ ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_276_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_277_image__image,axiom,
! [F: nat > nat,G2: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G2 @ X ) )
@ A ) ) ).
% image_image
thf(fact_278_imageE,axiom,
! [B3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_279_imageE,axiom,
! [B3: nat,F: fset_Bot_bot_term_a > nat,A: set_fs1788988886788723183term_a] :
( ( member_nat @ B3 @ ( image_8474690510679110544_a_nat @ F @ A ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2089387167371741008term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_280_imageE,axiom,
! [B3: nat,F: list_Bot_bot_term_a > nat,A: set_li5007820469839914319term_a] :
( ( member_nat @ B3 @ ( image_1574777119431997168_a_nat @ F @ A ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2850584719281785520term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_281_imageE,axiom,
! [B3: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,A: set_nat] :
( ( member2089387167371741008term_a @ B3 @ ( image_1785693973830655120term_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_282_imageE,axiom,
! [B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ B3 @ ( image_5390476545841808249term_a @ F @ A ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2089387167371741008term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_283_imageE,axiom,
! [B3: fset_Bot_bot_term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( member2089387167371741008term_a @ B3 @ ( image_4385135741129320153term_a @ F @ A ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2850584719281785520term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_284_imageE,axiom,
! [B3: list_Bot_bot_term_a,F: nat > list_Bot_bot_term_a,A: set_nat] :
( ( member2850584719281785520term_a @ B3 @ ( image_2546891525740699632term_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_285_imageE,axiom,
! [B3: list_Bot_bot_term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( member2850584719281785520term_a @ B3 @ ( image_6151674097751852761term_a @ F @ A ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2089387167371741008term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_286_imageE,axiom,
! [B3: list_Bot_bot_term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ B3 @ ( image_5146333293039364665term_a @ F @ A ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member2850584719281785520term_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_287_Collect__disj__eq,axiom,
! [P: list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > $o] :
( ( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su671802143010907771term_a @ ( collec4020393894392429550term_a @ P ) @ ( collec4020393894392429550term_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_288_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_289_Un__def,axiom,
( sup_su6676342596814492443term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A5 )
| ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_290_Un__def,axiom,
( sup_su671802143010907771term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A5 )
| ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_291_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A5 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_292_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_293_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_294_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_295_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_296_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_297_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_298_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_299_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_300_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_301_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_302_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_303_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_304_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_305_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_306_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_307_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_308_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_309_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_310_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_311_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_312_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_313_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_314_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_315_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_316_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_317_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_318_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_319_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_320_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_321_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_322_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_323_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_324_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_325_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_326_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_327_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_328_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_329_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_330_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_331_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_332_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_333_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_334_sorted__wrt__true,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_335_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_336_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2089387167371741008term_a @ B3 @ ( image_1785693973830655120term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_337_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: list_Bot_bot_term_a,F: nat > list_Bot_bot_term_a] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2850584719281785520term_a @ B3 @ ( image_2546891525740699632term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B3: nat,F: fset_Bot_bot_term_a > nat] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_8474690510679110544_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_339_rev__image__eqI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2089387167371741008term_a @ B3 @ ( image_5390476545841808249term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_340_rev__image__eqI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B3: list_Bot_bot_term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2850584719281785520term_a @ B3 @ ( image_6151674097751852761term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_341_rev__image__eqI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B3: nat,F: list_Bot_bot_term_a > nat] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_1574777119431997168_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_342_rev__image__eqI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B3: fset_Bot_bot_term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2089387167371741008term_a @ B3 @ ( image_4385135741129320153term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_343_rev__image__eqI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B3: list_Bot_bot_term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member2850584719281785520term_a @ B3 @ ( image_5146333293039364665term_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_344_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_345_image__cong,axiom,
! [M3: set_nat,N5: set_nat,F: nat > nat,G2: nat > nat] :
( ( M3 = N5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G2 @ N5 ) ) ) ) ).
% image_cong
thf(fact_346_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_347_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_348_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_349_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_350_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > fset_Bot_bot_term_a] :
( ( member_nat @ X2 @ A )
=> ( member2089387167371741008term_a @ ( F @ X2 ) @ ( image_1785693973830655120term_a @ F @ A ) ) ) ).
% imageI
thf(fact_351_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > list_Bot_bot_term_a] :
( ( member_nat @ X2 @ A )
=> ( member2850584719281785520term_a @ ( F @ X2 ) @ ( image_2546891525740699632term_a @ F @ A ) ) ) ).
% imageI
thf(fact_352_imageI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > nat] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_8474690510679110544_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_353_imageI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( member2089387167371741008term_a @ ( F @ X2 ) @ ( image_5390476545841808249term_a @ F @ A ) ) ) ).
% imageI
thf(fact_354_imageI,axiom,
! [X2: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X2 @ A )
=> ( member2850584719281785520term_a @ ( F @ X2 ) @ ( image_6151674097751852761term_a @ F @ A ) ) ) ).
% imageI
thf(fact_355_imageI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > nat] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_1574777119431997168_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_356_imageI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( member2089387167371741008term_a @ ( F @ X2 ) @ ( image_4385135741129320153term_a @ F @ A ) ) ) ).
% imageI
thf(fact_357_imageI,axiom,
! [X2: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X2 @ A )
=> ( member2850584719281785520term_a @ ( F @ X2 ) @ ( image_5146333293039364665term_a @ F @ A ) ) ) ).
% imageI
thf(fact_358_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_359_UnI2,axiom,
! [C: fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ B )
=> ( member2089387167371741008term_a @ C @ ( sup_su6676342596814492443term_a @ A @ B ) ) ) ).
% UnI2
thf(fact_360_UnI2,axiom,
! [C: list_Bot_bot_term_a,B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ B )
=> ( member2850584719281785520term_a @ C @ ( sup_su671802143010907771term_a @ A @ B ) ) ) ).
% UnI2
thf(fact_361_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_362_UnI1,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ A )
=> ( member2089387167371741008term_a @ C @ ( sup_su6676342596814492443term_a @ A @ B ) ) ) ).
% UnI1
thf(fact_363_UnI1,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ A )
=> ( member2850584719281785520term_a @ C @ ( sup_su671802143010907771term_a @ A @ B ) ) ) ).
% UnI1
thf(fact_364_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_365_UnE,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( sup_su6676342596814492443term_a @ A @ B ) )
=> ( ~ ( member2089387167371741008term_a @ C @ A )
=> ( member2089387167371741008term_a @ C @ B ) ) ) ).
% UnE
thf(fact_366_UnE,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( sup_su671802143010907771term_a @ A @ B ) )
=> ( ~ ( member2850584719281785520term_a @ C @ A )
=> ( member2850584719281785520term_a @ C @ B ) ) ) ).
% UnE
thf(fact_367_strict__sorted__equal,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_368_strict__sorted__imp__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_369_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_370_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_371_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_372_sorted__wrt__mono__rel,axiom,
! [Xs2: list_f3888513508467368777term_a,P: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o,Q: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ( ( member2089387167371741008term_a @ Y2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) ) ) )
=> ( ( sorted283000622152637068term_a @ P @ Xs2 )
=> ( sorted283000622152637068term_a @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_373_sorted__wrt__mono__rel,axiom,
! [Xs2: list_l7107345091518559913term_a,P: list_Bot_bot_term_a > list_Bot_bot_term_a > $o,Q: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ! [X3: list_Bot_bot_term_a,Y2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ( ( member2850584719281785520term_a @ Y2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) ) ) )
=> ( ( sorted1044198174062681580term_a @ P @ Xs2 )
=> ( sorted1044198174062681580term_a @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_374_sorted__wrt__mono__rel,axiom,
! [Xs2: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Xs2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs2 )
=> ( sorted_wrt_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_375_list__all2__lengthD,axiom,
! [P: bot_bot_term_a > bot_bot_term_a > $o,Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ( list_a6841061213177516955term_a @ P @ Xs2 @ Ys )
=> ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_s1103687553077312429term_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_376_list__all2__lengthD,axiom,
! [P: bot_bot_term_a > nat > $o,Xs2: list_Bot_bot_term_a,Ys: list_nat] :
( ( list_a782220476339921762_a_nat @ P @ Xs2 @ Ys )
=> ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_377_list__all2__lengthD,axiom,
! [P: nat > bot_bot_term_a > $o,Xs2: list_nat,Ys: list_Bot_bot_term_a] :
( ( list_a8962304646142687458term_a @ P @ Xs2 @ Ys )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s1103687553077312429term_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_378_list__all2__lengthD,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_379_list__all2__in__set2,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,Y: nat] :
( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_380_list__all2__in__set2,axiom,
! [P: fset_Bot_bot_term_a > nat > $o,Xs2: list_f3888513508467368777term_a,Ys: list_nat,Y: nat] :
( ( list_a4880437193775372818_a_nat @ P @ Xs2 @ Ys )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_381_list__all2__in__set2,axiom,
! [P: list_Bot_bot_term_a > nat > $o,Xs2: list_l7107345091518559913term_a,Ys: list_nat,Y: nat] :
( ( list_a7203895839383035250_a_nat @ P @ Xs2 @ Ys )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_382_list__all2__in__set2,axiom,
! [P: nat > fset_Bot_bot_term_a > $o,Xs2: list_nat,Ys: list_f3888513508467368777term_a,Y: fset_Bot_bot_term_a] :
( ( list_a7414812693781693202term_a @ P @ Xs2 @ Ys )
=> ( ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ Ys ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_383_list__all2__in__set2,axiom,
! [P: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o,Xs2: list_f3888513508467368777term_a,Ys: list_f3888513508467368777term_a,Y: fset_Bot_bot_term_a] :
( ( list_a8285269901158863867term_a @ P @ Xs2 @ Ys )
=> ( ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ Ys ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_384_list__all2__in__set2,axiom,
! [P: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o,Xs2: list_l7107345091518559913term_a,Ys: list_f3888513508467368777term_a,Y: fset_Bot_bot_term_a] :
( ( list_a7279929096446375771term_a @ P @ Xs2 @ Ys )
=> ( ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ Ys ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_385_list__all2__in__set2,axiom,
! [P: nat > list_Bot_bot_term_a > $o,Xs2: list_nat,Ys: list_l7107345091518559913term_a,Y: list_Bot_bot_term_a] :
( ( list_a8176010245691737714term_a @ P @ Xs2 @ Ys )
=> ( ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ Ys ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_386_list__all2__in__set2,axiom,
! [P: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o,Xs2: list_f3888513508467368777term_a,Ys: list_l7107345091518559913term_a,Y: list_Bot_bot_term_a] :
( ( list_a9046467453068908379term_a @ P @ Xs2 @ Ys )
=> ( ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ Ys ) )
=> ~ ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_387_list__all2__in__set2,axiom,
! [P: list_Bot_bot_term_a > list_Bot_bot_term_a > $o,Xs2: list_l7107345091518559913term_a,Ys: list_l7107345091518559913term_a,Y: list_Bot_bot_term_a] :
( ( list_a8041126648356420283term_a @ P @ Xs2 @ Ys )
=> ( ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ Ys ) )
=> ~ ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ~ ( P @ X3 @ Y ) ) ) ) ).
% list_all2_in_set2
thf(fact_388_list_Orel__cong,axiom,
! [X2: list_nat,Ya: list_nat,Y: list_nat,Xa2: list_nat,R: nat > nat > $o,Ra: nat > nat > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2_nat_nat @ R @ X2 @ Y )
= ( list_all2_nat_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_389_list_Orel__cong,axiom,
! [X2: list_nat,Ya: list_nat,Y: list_f3888513508467368777term_a,Xa2: list_f3888513508467368777term_a,R: nat > fset_Bot_bot_term_a > $o,Ra: nat > fset_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: nat,Yb: fset_Bot_bot_term_a] :
( ( member_nat @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a7414812693781693202term_a @ R @ X2 @ Y )
= ( list_a7414812693781693202term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_390_list_Orel__cong,axiom,
! [X2: list_nat,Ya: list_nat,Y: list_l7107345091518559913term_a,Xa2: list_l7107345091518559913term_a,R: nat > list_Bot_bot_term_a > $o,Ra: nat > list_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: nat,Yb: list_Bot_bot_term_a] :
( ( member_nat @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a8176010245691737714term_a @ R @ X2 @ Y )
= ( list_a8176010245691737714term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_391_list_Orel__cong,axiom,
! [X2: list_f3888513508467368777term_a,Ya: list_f3888513508467368777term_a,Y: list_nat,Xa2: list_nat,R: fset_Bot_bot_term_a > nat > $o,Ra: fset_Bot_bot_term_a > nat > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: nat] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a4880437193775372818_a_nat @ R @ X2 @ Y )
= ( list_a4880437193775372818_a_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_392_list_Orel__cong,axiom,
! [X2: list_f3888513508467368777term_a,Ya: list_f3888513508467368777term_a,Y: list_f3888513508467368777term_a,Xa2: list_f3888513508467368777term_a,R: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o,Ra: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ Ya ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a8285269901158863867term_a @ R @ X2 @ Y )
= ( list_a8285269901158863867term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_393_list_Orel__cong,axiom,
! [X2: list_f3888513508467368777term_a,Ya: list_f3888513508467368777term_a,Y: list_l7107345091518559913term_a,Xa2: list_l7107345091518559913term_a,R: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o,Ra: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: list_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ Ya ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a9046467453068908379term_a @ R @ X2 @ Y )
= ( list_a9046467453068908379term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_394_list_Orel__cong,axiom,
! [X2: list_l7107345091518559913term_a,Ya: list_l7107345091518559913term_a,Y: list_nat,Xa2: list_nat,R: list_Bot_bot_term_a > nat > $o,Ra: list_Bot_bot_term_a > nat > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: nat] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a7203895839383035250_a_nat @ R @ X2 @ Y )
= ( list_a7203895839383035250_a_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_395_list_Orel__cong,axiom,
! [X2: list_l7107345091518559913term_a,Ya: list_l7107345091518559913term_a,Y: list_f3888513508467368777term_a,Xa2: list_f3888513508467368777term_a,R: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o,Ra: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: fset_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ Ya ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a7279929096446375771term_a @ R @ X2 @ Y )
= ( list_a7279929096446375771term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_396_list_Orel__cong,axiom,
! [X2: list_l7107345091518559913term_a,Ya: list_l7107345091518559913term_a,Y: list_l7107345091518559913term_a,Xa2: list_l7107345091518559913term_a,R: list_Bot_bot_term_a > list_Bot_bot_term_a > $o,Ra: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ Ya ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Xa2 ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_a8041126648356420283term_a @ R @ X2 @ Y )
= ( list_a8041126648356420283term_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_397_list_Orel__mono__strong,axiom,
! [R: nat > nat > $o,X2: list_nat,Y: list_nat,Ra: nat > nat > $o] :
( ( list_all2_nat_nat @ R @ X2 @ Y )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2_nat_nat @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_398_list_Orel__mono__strong,axiom,
! [R: nat > fset_Bot_bot_term_a > $o,X2: list_nat,Y: list_f3888513508467368777term_a,Ra: nat > fset_Bot_bot_term_a > $o] :
( ( list_a7414812693781693202term_a @ R @ X2 @ Y )
=> ( ! [Z4: nat,Yb: fset_Bot_bot_term_a] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a7414812693781693202term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_399_list_Orel__mono__strong,axiom,
! [R: nat > list_Bot_bot_term_a > $o,X2: list_nat,Y: list_l7107345091518559913term_a,Ra: nat > list_Bot_bot_term_a > $o] :
( ( list_a8176010245691737714term_a @ R @ X2 @ Y )
=> ( ! [Z4: nat,Yb: list_Bot_bot_term_a] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a8176010245691737714term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_400_list_Orel__mono__strong,axiom,
! [R: fset_Bot_bot_term_a > nat > $o,X2: list_f3888513508467368777term_a,Y: list_nat,Ra: fset_Bot_bot_term_a > nat > $o] :
( ( list_a4880437193775372818_a_nat @ R @ X2 @ Y )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: nat] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ X2 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a4880437193775372818_a_nat @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_401_list_Orel__mono__strong,axiom,
! [R: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o,X2: list_f3888513508467368777term_a,Y: list_f3888513508467368777term_a,Ra: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( list_a8285269901158863867term_a @ R @ X2 @ Y )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ X2 ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a8285269901158863867term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_402_list_Orel__mono__strong,axiom,
! [R: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o,X2: list_f3888513508467368777term_a,Y: list_l7107345091518559913term_a,Ra: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( list_a9046467453068908379term_a @ R @ X2 @ Y )
=> ( ! [Z4: fset_Bot_bot_term_a,Yb: list_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ X2 ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a9046467453068908379term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_403_list_Orel__mono__strong,axiom,
! [R: list_Bot_bot_term_a > nat > $o,X2: list_l7107345091518559913term_a,Y: list_nat,Ra: list_Bot_bot_term_a > nat > $o] :
( ( list_a7203895839383035250_a_nat @ R @ X2 @ Y )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: nat] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ X2 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a7203895839383035250_a_nat @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_404_list_Orel__mono__strong,axiom,
! [R: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o,X2: list_l7107345091518559913term_a,Y: list_f3888513508467368777term_a,Ra: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( list_a7279929096446375771term_a @ R @ X2 @ Y )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: fset_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ X2 ) )
=> ( ( member2089387167371741008term_a @ Yb @ ( set_fs4478461793750015460term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a7279929096446375771term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_405_list_Orel__mono__strong,axiom,
! [R: list_Bot_bot_term_a > list_Bot_bot_term_a > $o,X2: list_l7107345091518559913term_a,Y: list_l7107345091518559913term_a,Ra: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( list_a8041126648356420283term_a @ R @ X2 @ Y )
=> ( ! [Z4: list_Bot_bot_term_a,Yb: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ X2 ) )
=> ( ( member2850584719281785520term_a @ Yb @ ( set_li5239659345660059972term_a @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_a8041126648356420283term_a @ Ra @ X2 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_406_list_Orel__refl__strong,axiom,
! [X2: list_nat,Ra: nat > nat > $o] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_all2_nat_nat @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_407_list_Orel__refl__strong,axiom,
! [X2: list_f3888513508467368777term_a,Ra: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ! [Z4: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Z4 @ ( set_fs4478461793750015460term_a @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_a8285269901158863867term_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_408_list_Orel__refl__strong,axiom,
! [X2: list_l7107345091518559913term_a,Ra: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ! [Z4: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Z4 @ ( set_li5239659345660059972term_a @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_a8041126648356420283term_a @ Ra @ X2 @ X2 ) ) ).
% list.rel_refl_strong
thf(fact_409_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_410_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_411_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_412_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_413_image__subsetI,axiom,
! [A: set_nat,F: nat > fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member2089387167371741008term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le6553944718276964943term_a @ ( image_1785693973830655120term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_414_image__subsetI,axiom,
! [A: set_nat,F: nat > list_Bot_bot_term_a,B: set_li5007820469839914319term_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member2850584719281785520term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le549404264473380271term_a @ ( image_2546891525740699632term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_415_image__subsetI,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > nat,B: set_nat] :
( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_8474690510679110544_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_416_image__subsetI,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ( member2089387167371741008term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le6553944718276964943term_a @ ( image_5390476545841808249term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_417_image__subsetI,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a,B: set_li5007820469839914319term_a] :
( ! [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
=> ( member2850584719281785520term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le549404264473380271term_a @ ( image_6151674097751852761term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_418_image__subsetI,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > nat,B: set_nat] :
( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_1574777119431997168_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_419_image__subsetI,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ( member2089387167371741008term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le6553944718276964943term_a @ ( image_4385135741129320153term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_420_image__subsetI,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a,B: set_li5007820469839914319term_a] :
( ! [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
=> ( member2850584719281785520term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le549404264473380271term_a @ ( image_5146333293039364665term_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_421_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_422_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_423_infinite__Un,axiom,
! [S: set_li5007820469839914319term_a,T: set_li5007820469839914319term_a] :
( ( ~ ( finite491101672212324272term_a @ ( sup_su671802143010907771term_a @ S @ T ) ) )
= ( ~ ( finite491101672212324272term_a @ S )
| ~ ( finite491101672212324272term_a @ T ) ) ) ).
% infinite_Un
thf(fact_424_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_425_infinite__Un,axiom,
! [S: set_fs1788988886788723183term_a,T: set_fs1788988886788723183term_a] :
( ( ~ ( finite8953276157157055568term_a @ ( sup_su6676342596814492443term_a @ S @ T ) ) )
= ( ~ ( finite8953276157157055568term_a @ S )
| ~ ( finite8953276157157055568term_a @ T ) ) ) ).
% infinite_Un
thf(fact_426_Un__infinite,axiom,
! [S: set_li5007820469839914319term_a,T: set_li5007820469839914319term_a] :
( ~ ( finite491101672212324272term_a @ S )
=> ~ ( finite491101672212324272term_a @ ( sup_su671802143010907771term_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_427_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_428_Un__infinite,axiom,
! [S: set_fs1788988886788723183term_a,T: set_fs1788988886788723183term_a] :
( ~ ( finite8953276157157055568term_a @ S )
=> ~ ( finite8953276157157055568term_a @ ( sup_su6676342596814492443term_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_429_finite__UnI,axiom,
! [F2: set_li5007820469839914319term_a,G: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ F2 )
=> ( ( finite491101672212324272term_a @ G )
=> ( finite491101672212324272term_a @ ( sup_su671802143010907771term_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_430_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_431_finite__UnI,axiom,
! [F2: set_fs1788988886788723183term_a,G: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ F2 )
=> ( ( finite8953276157157055568term_a @ G )
=> ( finite8953276157157055568term_a @ ( sup_su6676342596814492443term_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_432_order__le__imp__less__or__eq,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( ord_le1098638842511489549term_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_433_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_434_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_435_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le1098638842511489549term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_436_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_437_order__less__le__subst1,axiom,
! [A2: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_438_order__less__le__subst1,axiom,
! [A2: nat,F: fset_Bot_bot_term_a > nat,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_439_order__less__le__subst1,axiom,
! [A2: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,B3: nat,C: nat] :
( ( ord_le1098638842511489549term_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_440_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_441_order__le__less__subst2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_le1098638842511489549term_a @ ( F @ B3 ) @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_442_order__le__less__subst2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,F: fset_Bot_bot_term_a > nat,C: nat] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_443_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le1098638842511489549term_a @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7216997114146882585term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_444_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_445_order__le__less__subst1,axiom,
! [A2: fset_Bot_bot_term_a,F: nat > fset_Bot_bot_term_a,B3: nat,C: nat] :
( ( ord_le7216997114146882585term_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le1098638842511489549term_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1098638842511489549term_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_446_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_447_order__less__le__trans,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a,Z2: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ X2 @ Y )
=> ( ( ord_le7216997114146882585term_a @ Y @ Z2 )
=> ( ord_le1098638842511489549term_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_448_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_449_order__le__less__trans,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a,Z2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( ord_le1098638842511489549term_a @ Y @ Z2 )
=> ( ord_le1098638842511489549term_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_450_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_451_order__neq__le__trans,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( A2 != B3 )
=> ( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ord_le1098638842511489549term_a @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_452_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_453_order__le__neq__trans,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le1098638842511489549term_a @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_454_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_455_order__less__imp__le,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ X2 @ Y )
=> ( ord_le7216997114146882585term_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_456_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_457_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_458_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_459_order__less__le,axiom,
( ord_le1098638842511489549term_a
= ( ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_460_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_461_order__le__less,axiom,
( ord_le7216997114146882585term_a
= ( ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_462_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_463_dual__order_Ostrict__implies__order,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ B3 @ A2 )
=> ( ord_le7216997114146882585term_a @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_464_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_465_order_Ostrict__implies__order,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A2 @ B3 )
=> ( ord_le7216997114146882585term_a @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_466_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_467_dual__order_Ostrict__iff__not,axiom,
( ord_le1098638842511489549term_a
= ( ^ [B4: fset_Bot_bot_term_a,A3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B4 @ A3 )
& ~ ( ord_le7216997114146882585term_a @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_468_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_469_dual__order_Ostrict__trans2,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ B3 @ A2 )
=> ( ( ord_le7216997114146882585term_a @ C @ B3 )
=> ( ord_le1098638842511489549term_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_470_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_471_dual__order_Ostrict__trans1,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( ( ord_le1098638842511489549term_a @ C @ B3 )
=> ( ord_le1098638842511489549term_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_472_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_473_dual__order_Ostrict__iff__order,axiom,
( ord_le1098638842511489549term_a
= ( ^ [B4: fset_Bot_bot_term_a,A3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_474_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_475_dual__order_Oorder__iff__strict,axiom,
( ord_le7216997114146882585term_a
= ( ^ [B4: fset_Bot_bot_term_a,A3: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_476_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_477_order_Ostrict__iff__not,axiom,
( ord_le1098638842511489549term_a
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A3 @ B4 )
& ~ ( ord_le7216997114146882585term_a @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_478_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_479_order_Ostrict__trans2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A2 @ B3 )
=> ( ( ord_le7216997114146882585term_a @ B3 @ C )
=> ( ord_le1098638842511489549term_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_480_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_481_order_Ostrict__trans1,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( ord_le1098638842511489549term_a @ B3 @ C )
=> ( ord_le1098638842511489549term_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_482_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_483_order_Ostrict__iff__order,axiom,
( ord_le1098638842511489549term_a
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_484_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_485_order_Oorder__iff__strict,axiom,
( ord_le7216997114146882585term_a
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_486_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_487_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_488_less__le__not__le,axiom,
( ord_le1098638842511489549term_a
= ( ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X @ Y4 )
& ~ ( ord_le7216997114146882585term_a @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_489_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_490_antisym__conv2,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( ~ ( ord_le1098638842511489549term_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_491_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_492_antisym__conv1,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ~ ( ord_le1098638842511489549term_a @ X2 @ Y )
=> ( ( ord_le7216997114146882585term_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_493_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_494_nless__le,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ~ ( ord_le1098638842511489549term_a @ A2 @ B3 ) )
= ( ~ ( ord_le7216997114146882585term_a @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_495_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_496_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_497_leD,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Y @ X2 )
=> ~ ( ord_le1098638842511489549term_a @ X2 @ Y ) ) ).
% leD
thf(fact_498_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_499_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_500_length__induct,axiom,
! [P: list_Bot_bot_term_a > $o,Xs2: list_Bot_bot_term_a] :
( ! [Xs: list_Bot_bot_term_a] :
( ! [Ys3: list_Bot_bot_term_a] :
( ( ord_less_nat @ ( size_s1103687553077312429term_a @ Ys3 ) @ ( size_s1103687553077312429term_a @ Xs ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_501_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_502_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_503_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_504_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_505_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_506_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_507_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_508_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( F @ K2 @ I3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_509_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_510_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_511_pigeonhole__infinite,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite491101672212324272term_a @ ( image_5146333293039364665term_a @ F @ A ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_512_pigeonhole__infinite,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > nat] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite_finite_nat @ ( image_1574777119431997168_a_nat @ F @ A ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_513_pigeonhole__infinite,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite8953276157157055568term_a @ ( image_4385135741129320153term_a @ F @ A ) )
=> ? [X3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X3 @ A )
& ~ ( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [A3: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_514_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > list_Bot_bot_term_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite491101672212324272term_a @ ( image_2546891525740699632term_a @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_515_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
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_516_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > fset_Bot_bot_term_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite8953276157157055568term_a @ ( image_1785693973830655120term_a @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_517_pigeonhole__infinite,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite491101672212324272term_a @ ( image_6151674097751852761term_a @ F @ A ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_518_pigeonhole__infinite,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > nat] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite_finite_nat @ ( image_8474690510679110544_a_nat @ F @ A ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_519_pigeonhole__infinite,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite8953276157157055568term_a @ ( image_5390476545841808249term_a @ F @ A ) )
=> ? [X3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X3 @ A )
& ~ ( finite8953276157157055568term_a
@ ( collec3259196342482385038term_a
@ ^ [A3: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A3 @ A )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_520_image__Collect__subsetI,axiom,
! [P: list_Bot_bot_term_a > $o,F: list_Bot_bot_term_a > nat,B: set_nat] :
( ! [X3: list_Bot_bot_term_a] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_1574777119431997168_a_nat @ F @ ( collec4020393894392429550term_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_521_image__Collect__subsetI,axiom,
! [P: list_Bot_bot_term_a > $o,F: list_Bot_bot_term_a > fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: list_Bot_bot_term_a] :
( ( P @ X3 )
=> ( member2089387167371741008term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le6553944718276964943term_a @ ( image_4385135741129320153term_a @ F @ ( collec4020393894392429550term_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_522_image__Collect__subsetI,axiom,
! [P: list_Bot_bot_term_a > $o,F: list_Bot_bot_term_a > list_Bot_bot_term_a,B: set_li5007820469839914319term_a] :
( ! [X3: list_Bot_bot_term_a] :
( ( P @ X3 )
=> ( member2850584719281785520term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le549404264473380271term_a @ ( image_5146333293039364665term_a @ F @ ( collec4020393894392429550term_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_523_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_524_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > fset_Bot_bot_term_a,B: set_fs1788988886788723183term_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member2089387167371741008term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le6553944718276964943term_a @ ( image_1785693973830655120term_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_525_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > list_Bot_bot_term_a,B: set_li5007820469839914319term_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member2850584719281785520term_a @ ( F @ X3 ) @ B ) )
=> ( ord_le549404264473380271term_a @ ( image_2546891525740699632term_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_526_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_527_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( P @ K5 )
& ( ord_less_nat @ K5 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_528_finite__surj,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( finite491101672212324272term_a @ A )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_5146333293039364665term_a @ F @ A ) )
=> ( finite491101672212324272term_a @ B ) ) ) ).
% finite_surj
thf(fact_529_finite__surj,axiom,
! [A: set_li5007820469839914319term_a,B: set_nat,F: list_Bot_bot_term_a > nat] :
( ( finite491101672212324272term_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1574777119431997168_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_530_finite__surj,axiom,
! [A: set_li5007820469839914319term_a,B: set_fs1788988886788723183term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( finite491101672212324272term_a @ A )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_4385135741129320153term_a @ F @ A ) )
=> ( finite8953276157157055568term_a @ B ) ) ) ).
% finite_surj
thf(fact_531_finite__surj,axiom,
! [A: set_nat,B: set_li5007820469839914319term_a,F: nat > list_Bot_bot_term_a] :
( ( finite_finite_nat @ A )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_2546891525740699632term_a @ F @ A ) )
=> ( finite491101672212324272term_a @ B ) ) ) ).
% finite_surj
thf(fact_532_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_533_finite__surj,axiom,
! [A: set_nat,B: set_fs1788988886788723183term_a,F: nat > fset_Bot_bot_term_a] :
( ( finite_finite_nat @ A )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_1785693973830655120term_a @ F @ A ) )
=> ( finite8953276157157055568term_a @ B ) ) ) ).
% finite_surj
thf(fact_534_finite__surj,axiom,
! [A: set_fs1788988886788723183term_a,B: set_li5007820469839914319term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_6151674097751852761term_a @ F @ A ) )
=> ( finite491101672212324272term_a @ B ) ) ) ).
% finite_surj
thf(fact_535_finite__surj,axiom,
! [A: set_fs1788988886788723183term_a,B: set_nat,F: fset_Bot_bot_term_a > nat] :
( ( finite8953276157157055568term_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_8474690510679110544_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_536_finite__surj,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_5390476545841808249term_a @ F @ A ) )
=> ( finite8953276157157055568term_a @ B ) ) ) ).
% finite_surj
thf(fact_537_finite__subset__image,axiom,
! [B: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > list_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_5146333293039364665term_a @ F @ A ) )
=> ? [C3: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ C3 @ A )
& ( finite491101672212324272term_a @ C3 )
& ( B
= ( image_5146333293039364665term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_538_finite__subset__image,axiom,
! [B: set_li5007820469839914319term_a,F: nat > list_Bot_bot_term_a,A: set_nat] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_2546891525740699632term_a @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_2546891525740699632term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_539_finite__subset__image,axiom,
! [B: set_li5007820469839914319term_a,F: fset_Bot_bot_term_a > list_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ B @ ( image_6151674097751852761term_a @ F @ A ) )
=> ? [C3: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ C3 @ A )
& ( finite8953276157157055568term_a @ C3 )
& ( B
= ( image_6151674097751852761term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_540_finite__subset__image,axiom,
! [B: set_nat,F: list_Bot_bot_term_a > nat,A: set_li5007820469839914319term_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1574777119431997168_a_nat @ F @ A ) )
=> ? [C3: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ C3 @ A )
& ( finite491101672212324272term_a @ C3 )
& ( B
= ( image_1574777119431997168_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_541_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 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_542_finite__subset__image,axiom,
! [B: set_nat,F: fset_Bot_bot_term_a > nat,A: set_fs1788988886788723183term_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_8474690510679110544_a_nat @ F @ A ) )
=> ? [C3: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ C3 @ A )
& ( finite8953276157157055568term_a @ C3 )
& ( B
= ( image_8474690510679110544_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_543_finite__subset__image,axiom,
! [B: set_fs1788988886788723183term_a,F: list_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_li5007820469839914319term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_4385135741129320153term_a @ F @ A ) )
=> ? [C3: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ C3 @ A )
& ( finite491101672212324272term_a @ C3 )
& ( B
= ( image_4385135741129320153term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_544_finite__subset__image,axiom,
! [B: set_fs1788988886788723183term_a,F: nat > fset_Bot_bot_term_a,A: set_nat] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_1785693973830655120term_a @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_1785693973830655120term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_545_finite__subset__image,axiom,
! [B: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ B @ ( image_5390476545841808249term_a @ F @ A ) )
=> ? [C3: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ C3 @ A )
& ( finite8953276157157055568term_a @ C3 )
& ( B
= ( image_5390476545841808249term_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_546_ex__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > list_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: set_li5007820469839914319term_a > $o] :
( ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_5146333293039364665term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A )
& ( P @ ( image_5146333293039364665term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_547_ex__finite__subset__image,axiom,
! [F: nat > list_Bot_bot_term_a,A: set_nat,P: set_li5007820469839914319term_a > $o] :
( ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_2546891525740699632term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_2546891525740699632term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_548_ex__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > list_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: set_li5007820469839914319term_a > $o] :
( ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_6151674097751852761term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A )
& ( P @ ( image_6151674097751852761term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_549_ex__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > nat,A: set_li5007820469839914319term_a,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_1574777119431997168_a_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A )
& ( P @ ( image_1574777119431997168_a_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_550_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_551_ex__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > nat,A: set_fs1788988886788723183term_a,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_8474690510679110544_a_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A )
& ( P @ ( image_8474690510679110544_a_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_552_ex__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: set_fs1788988886788723183term_a > $o] :
( ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_4385135741129320153term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A )
& ( P @ ( image_4385135741129320153term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_553_ex__finite__subset__image,axiom,
! [F: nat > fset_Bot_bot_term_a,A: set_nat,P: set_fs1788988886788723183term_a > $o] :
( ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_1785693973830655120term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_1785693973830655120term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_554_ex__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: set_fs1788988886788723183term_a > $o] :
( ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_5390476545841808249term_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A )
& ( P @ ( image_5390476545841808249term_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_555_all__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > list_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: set_li5007820469839914319term_a > $o] :
( ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_5146333293039364665term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A ) )
=> ( P @ ( image_5146333293039364665term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_556_all__finite__subset__image,axiom,
! [F: nat > list_Bot_bot_term_a,A: set_nat,P: set_li5007820469839914319term_a > $o] :
( ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_2546891525740699632term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_2546891525740699632term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_557_all__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > list_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: set_li5007820469839914319term_a > $o] :
( ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ ( image_6151674097751852761term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A ) )
=> ( P @ ( image_6151674097751852761term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_558_all__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > nat,A: set_li5007820469839914319term_a,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_1574777119431997168_a_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A ) )
=> ( P @ ( image_1574777119431997168_a_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_559_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_560_all__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > nat,A: set_fs1788988886788723183term_a,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_8474690510679110544_a_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A ) )
=> ( P @ ( image_8474690510679110544_a_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_561_all__finite__subset__image,axiom,
! [F: list_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_li5007820469839914319term_a,P: set_fs1788988886788723183term_a > $o] :
( ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_4385135741129320153term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_li5007820469839914319term_a] :
( ( ( finite491101672212324272term_a @ B2 )
& ( ord_le549404264473380271term_a @ B2 @ A ) )
=> ( P @ ( image_4385135741129320153term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_562_all__finite__subset__image,axiom,
! [F: nat > fset_Bot_bot_term_a,A: set_nat,P: set_fs1788988886788723183term_a > $o] :
( ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_1785693973830655120term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_1785693973830655120term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_563_all__finite__subset__image,axiom,
! [F: fset_Bot_bot_term_a > fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,P: set_fs1788988886788723183term_a > $o] :
( ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ ( image_5390476545841808249term_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_fs1788988886788723183term_a] :
( ( ( finite8953276157157055568term_a @ B2 )
& ( ord_le6553944718276964943term_a @ B2 @ A ) )
=> ( P @ ( image_5390476545841808249term_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_564_remdups__sort_I1_J,axiom,
! [Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( missin6101193410121742181rt_nat @ Xs2 ) ) ).
% remdups_sort(1)
thf(fact_565_finite__maxlen,axiom,
! [M3: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ M3 )
=> ? [N3: nat] :
! [X6: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X6 @ M3 )
=> ( ord_less_nat @ ( size_s1103687553077312429term_a @ X6 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_566_finite__maxlen,axiom,
! [M3: set_list_nat] :
( ( finite8100373058378681591st_nat @ M3 )
=> ? [N3: nat] :
! [X6: list_nat] :
( ( member_list_nat @ X6 @ M3 )
=> ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_567_sorted__distinct__set__unique,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( distinct_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs2 )
= ( set_nat2 @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_568_sup_Obounded__iff,axiom,
! [B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ B3 @ C ) @ A2 )
= ( ( ord_le7216997114146882585term_a @ B3 @ A2 )
& ( ord_le7216997114146882585term_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_569_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_570_le__sup__iff,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a,Z2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ X2 @ Y ) @ Z2 )
= ( ( ord_le7216997114146882585term_a @ X2 @ Z2 )
& ( ord_le7216997114146882585term_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_571_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_572_finite__psubset__induct,axiom,
! [A: set_li5007820469839914319term_a,P: set_li5007820469839914319term_a > $o] :
( ( finite491101672212324272term_a @ A )
=> ( ! [A7: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A7 )
=> ( ! [B7: set_li5007820469839914319term_a] :
( ( ord_le4209214868046015395term_a @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_573_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_574_finite__psubset__induct,axiom,
! [A: set_fs1788988886788723183term_a,P: set_fs1788988886788723183term_a > $o] :
( ( finite8953276157157055568term_a @ A )
=> ( ! [A7: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A7 )
=> ( ! [B7: set_fs1788988886788723183term_a] :
( ( ord_le990383284994824259term_a @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_575_funion__mono,axiom,
! [A: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,D: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ C2 )
=> ( ( ord_le7216997114146882585term_a @ B @ D )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ A @ B ) @ ( sup_su6230847479081161957term_a @ C2 @ D ) ) ) ) ).
% funion_mono
thf(fact_576_funion__least,axiom,
! [A: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ C2 )
=> ( ( ord_le7216997114146882585term_a @ B @ C2 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ A @ B ) @ C2 ) ) ) ).
% funion_least
thf(fact_577_funion__upper1,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ A @ ( sup_su6230847479081161957term_a @ A @ B ) ) ).
% funion_upper1
thf(fact_578_funion__upper2,axiom,
! [B: fset_Bot_bot_term_a,A: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ B @ ( sup_su6230847479081161957term_a @ A @ B ) ) ).
% funion_upper2
thf(fact_579_funion__absorb1,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( sup_su6230847479081161957term_a @ A @ B )
= B ) ) ).
% funion_absorb1
thf(fact_580_funion__absorb2,axiom,
! [B: fset_Bot_bot_term_a,A: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B @ A )
=> ( ( sup_su6230847479081161957term_a @ A @ B )
= A ) ) ).
% funion_absorb2
thf(fact_581_fsubset__funion__eq,axiom,
( ord_le7216997114146882585term_a
= ( ^ [A5: fset_Bot_bot_term_a,B2: fset_Bot_bot_term_a] :
( ( sup_su6230847479081161957term_a @ A5 @ B2 )
= B2 ) ) ) ).
% fsubset_funion_eq
thf(fact_582_pfsubset__eq,axiom,
( ord_le1098638842511489549term_a
= ( ^ [A5: fset_Bot_bot_term_a,B2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% pfsubset_eq
thf(fact_583_less__fset__def,axiom,
( ord_le1098638842511489549term_a
= ( ^ [Xs3: fset_Bot_bot_term_a,Ys2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Xs3 @ Ys2 )
& ( Xs3 != Ys2 ) ) ) ) ).
% less_fset_def
thf(fact_584_pfsubset__imp__fsubset,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A @ B )
=> ( ord_le7216997114146882585term_a @ A @ B ) ) ).
% pfsubset_imp_fsubset
thf(fact_585_fsubset__not__fsubset__eq,axiom,
( ord_le1098638842511489549term_a
= ( ^ [A5: fset_Bot_bot_term_a,B2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A5 @ B2 )
& ~ ( ord_le7216997114146882585term_a @ B2 @ A5 ) ) ) ) ).
% fsubset_not_fsubset_eq
thf(fact_586_fsubset__pfsubset__trans,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( ord_le1098638842511489549term_a @ B @ C2 )
=> ( ord_le1098638842511489549term_a @ A @ C2 ) ) ) ).
% fsubset_pfsubset_trans
thf(fact_587_pfsubset__fsubset__trans,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A @ B )
=> ( ( ord_le7216997114146882585term_a @ B @ C2 )
=> ( ord_le1098638842511489549term_a @ A @ C2 ) ) ) ).
% pfsubset_fsubset_trans
thf(fact_588_fsubset__iff__pfsubset__eq,axiom,
( ord_le7216997114146882585term_a
= ( ^ [A5: fset_Bot_bot_term_a,B2: fset_Bot_bot_term_a] :
( ( ord_le1098638842511489549term_a @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% fsubset_iff_pfsubset_eq
thf(fact_589_sup__set__def,axiom,
( sup_su6676342596814492443term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( collec3259196342482385038term_a
@ ( sup_su5242116801355244330rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A5 )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_590_sup__set__def,axiom,
( sup_su671802143010907771term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( collec4020393894392429550term_a
@ ( sup_su7755517514950117322rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A5 )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_591_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_592_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_593_sup__Un__eq,axiom,
! [R: set_fs1788988886788723183term_a,S: set_fs1788988886788723183term_a] :
( ( sup_su5242116801355244330rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ R )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ S ) )
= ( ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ ( sup_su6676342596814492443term_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_594_sup__Un__eq,axiom,
! [R: set_li5007820469839914319term_a,S: set_li5007820469839914319term_a] :
( ( sup_su7755517514950117322rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ R )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ S ) )
= ( ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ ( sup_su671802143010907771term_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_595_nat__seg__image__imp__finite,axiom,
! [A: set_li5007820469839914319term_a,F: nat > list_Bot_bot_term_a,N: nat] :
( ( A
= ( image_2546891525740699632term_a @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) )
=> ( finite491101672212324272term_a @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_596_nat__seg__image__imp__finite,axiom,
! [A: set_nat,F: nat > nat,N: nat] :
( ( A
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) )
=> ( finite_finite_nat @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_597_nat__seg__image__imp__finite,axiom,
! [A: set_fs1788988886788723183term_a,F: nat > fset_Bot_bot_term_a,N: nat] :
( ( A
= ( image_1785693973830655120term_a @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) )
=> ( finite8953276157157055568term_a @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_598_finite__conv__nat__seg__image,axiom,
( finite491101672212324272term_a
= ( ^ [A5: set_li5007820469839914319term_a] :
? [N2: nat,F3: nat > list_Bot_bot_term_a] :
( A5
= ( image_2546891525740699632term_a @ F3
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_599_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
? [N2: nat,F3: nat > nat] :
( A5
= ( image_nat_nat @ F3
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_600_finite__conv__nat__seg__image,axiom,
( finite8953276157157055568term_a
= ( ^ [A5: set_fs1788988886788723183term_a] :
? [N2: nat,F3: nat > fset_Bot_bot_term_a] :
( A5
= ( image_1785693973830655120term_a @ F3
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_601_sup_OcoboundedI2,axiom,
! [C: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ C @ B3 )
=> ( ord_le7216997114146882585term_a @ C @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_602_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_603_sup_OcoboundedI1,axiom,
! [C: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ C @ A2 )
=> ( ord_le7216997114146882585term_a @ C @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_604_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_605_sup_Oabsorb__iff2,axiom,
( ord_le7216997114146882585term_a
= ( ^ [A3: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
( ( sup_su6230847479081161957term_a @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_606_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_607_sup_Oabsorb__iff1,axiom,
( ord_le7216997114146882585term_a
= ( ^ [B4: fset_Bot_bot_term_a,A3: fset_Bot_bot_term_a] :
( ( sup_su6230847479081161957term_a @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_608_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_609_sup_Ocobounded2,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ B3 @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_610_sup_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_611_sup_Ocobounded1,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ A2 @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_612_sup_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_613_sup_Oorder__iff,axiom,
( ord_le7216997114146882585term_a
= ( ^ [B4: fset_Bot_bot_term_a,A3: fset_Bot_bot_term_a] :
( A3
= ( sup_su6230847479081161957term_a @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_614_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_615_sup_OboundedI,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( ( ord_le7216997114146882585term_a @ C @ A2 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_616_sup_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_617_sup_OboundedE,axiom,
! [B3: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ~ ( ord_le7216997114146882585term_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_618_sup_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_619_sup__absorb2,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ Y )
=> ( ( sup_su6230847479081161957term_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_620_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_621_sup__absorb1,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Y @ X2 )
=> ( ( sup_su6230847479081161957term_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_622_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_623_sup_Oabsorb2,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ B3 )
=> ( ( sup_su6230847479081161957term_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_624_sup_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_625_sup_Oabsorb1,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( ( sup_su6230847479081161957term_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_626_sup_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_627_sup__unique,axiom,
! [F: fset_Bot_bot_term_a > fset_Bot_bot_term_a > fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] :
( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: fset_Bot_bot_term_a,Y2: fset_Bot_bot_term_a,Z4: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Y2 @ X3 )
=> ( ( ord_le7216997114146882585term_a @ Z4 @ X3 )
=> ( ord_le7216997114146882585term_a @ ( F @ Y2 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su6230847479081161957term_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_628_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_629_sup_OorderI,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( A2
= ( sup_su6230847479081161957term_a @ A2 @ B3 ) )
=> ( ord_le7216997114146882585term_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_630_sup_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_631_sup_OorderE,axiom,
! [B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ B3 @ A2 )
=> ( A2
= ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_632_sup_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_633_le__iff__sup,axiom,
( ord_le7216997114146882585term_a
= ( ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] :
( ( sup_su6230847479081161957term_a @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_634_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_635_sup__least,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a,Z2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ Y @ X2 )
=> ( ( ord_le7216997114146882585term_a @ Z2 @ X2 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_636_sup__least,axiom,
! [Y: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_637_sup__mono,axiom,
! [A2: fset_Bot_bot_term_a,C: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,D2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ C )
=> ( ( ord_le7216997114146882585term_a @ B3 @ D2 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) @ ( sup_su6230847479081161957term_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_638_sup__mono,axiom,
! [A2: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_639_sup_Omono,axiom,
! [C: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,D2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ C @ A2 )
=> ( ( ord_le7216997114146882585term_a @ D2 @ B3 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ C @ D2 ) @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_640_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_641_le__supI2,axiom,
! [X2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ B3 )
=> ( ord_le7216997114146882585term_a @ X2 @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_642_le__supI2,axiom,
! [X2: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_643_le__supI1,axiom,
! [X2: fset_Bot_bot_term_a,A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ X2 @ A2 )
=> ( ord_le7216997114146882585term_a @ X2 @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_644_le__supI1,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_645_sup__ge2,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ Y @ ( sup_su6230847479081161957term_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_646_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_647_sup__ge1,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ X2 @ ( sup_su6230847479081161957term_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_648_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_649_le__supI,axiom,
! [A2: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A2 @ X2 )
=> ( ( ord_le7216997114146882585term_a @ B3 @ X2 )
=> ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_650_le__supI,axiom,
! [A2: nat,X2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_651_le__supE,axiom,
! [A2: fset_Bot_bot_term_a,B3: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( sup_su6230847479081161957term_a @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_le7216997114146882585term_a @ A2 @ X2 )
=> ~ ( ord_le7216997114146882585term_a @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_652_le__supE,axiom,
! [A2: nat,B3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_653_inf__sup__ord_I3_J,axiom,
! [X2: fset_Bot_bot_term_a,Y: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ X2 @ ( sup_su6230847479081161957term_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_654_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_655_inf__sup__ord_I4_J,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ Y @ ( sup_su6230847479081161957term_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_656_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_657_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_658_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_659_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B4 ) )
& ( A3 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_660_sup_Ostrict__boundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_661_sup_Oabsorb4,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_662_sup_Oabsorb3,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb3
thf(fact_663_less__supI2,axiom,
! [X2: nat,B3: nat,A2: nat] :
( ( ord_less_nat @ X2 @ B3 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% less_supI2
thf(fact_664_less__supI1,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% less_supI1
thf(fact_665_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_666_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B3 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_667_pinf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% pinf(6)
thf(fact_668_pinf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% pinf(8)
thf(fact_669_minf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% minf(6)
thf(fact_670_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_set_def
thf(fact_671_less__set__def,axiom,
( ord_le990383284994824259term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( ord_le5011019433847422978rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A5 )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ).
% less_set_def
thf(fact_672_less__set__def,axiom,
( ord_le4209214868046015395term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( ord_le7524420147442295970rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A5 )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ).
% less_set_def
thf(fact_673_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_674_psubsetD,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a,C: fset_Bot_bot_term_a] :
( ( ord_le990383284994824259term_a @ A @ B )
=> ( ( member2089387167371741008term_a @ C @ A )
=> ( member2089387167371741008term_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_675_psubsetD,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a,C: list_Bot_bot_term_a] :
( ( ord_le4209214868046015395term_a @ A @ B )
=> ( ( member2850584719281785520term_a @ C @ A )
=> ( member2850584719281785520term_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_676_minf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T3 @ X6 ) ) ).
% minf(7)
thf(fact_677_minf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T3 ) ) ).
% minf(5)
thf(fact_678_minf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T3 ) ) ).
% minf(4)
thf(fact_679_minf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T3 ) ) ).
% minf(3)
thf(fact_680_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_681_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_682_pinf_I7_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T3 @ X6 ) ) ).
% pinf(7)
thf(fact_683_pinf_I5_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T3 ) ) ).
% pinf(5)
thf(fact_684_pinf_I4_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(4)
thf(fact_685_pinf_I3_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T3 ) ) ).
% pinf(3)
thf(fact_686_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_687_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_688_minf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% minf(8)
thf(fact_689_strict__sorted__equal__Uniq,axiom,
! [A: set_nat] :
( uniq_list_nat
@ ^ [Xs3: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs3 )
& ( ( set_nat2 @ Xs3 )
= A ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_690_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic5753546321084859743term_a @ sup_su6230847479081161957term_a
@ ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ Y4 @ X )
@ ^ [X: fset_Bot_bot_term_a,Y4: fset_Bot_bot_term_a] : ( ord_le1098638842511489549term_a @ Y4 @ X ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_691_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic6009151579333465974et_nat @ sup_sup_nat
@ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X )
@ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_692_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( distinct_nat @ Xs2 )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs2 ) )
= Xs2 ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_693_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_694_finite__Pow__iff,axiom,
! [A: set_li5007820469839914319term_a] :
( ( finite1423361352422162918term_a @ ( pow_li3987379423859365436term_a @ A ) )
= ( finite491101672212324272term_a @ A ) ) ).
% finite_Pow_iff
thf(fact_695_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_696_finite__Pow__iff,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( finite7427901806225747590term_a @ ( pow_fs3226181871949320924term_a @ A ) )
= ( finite8953276157157055568term_a @ A ) ) ).
% finite_Pow_iff
thf(fact_697_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_698_Powp__Pow__eq,axiom,
! [A: set_nat] :
( ( powp_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= ( ^ [X: set_nat] : ( member_set_nat @ X @ ( pow_nat @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_699_Powp__Pow__eq,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( powp_f4750820763841964122term_a
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A ) )
= ( ^ [X: set_fs1788988886788723183term_a] : ( member2313390693227633030term_a @ X @ ( pow_fs3226181871949320924term_a @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_700_Powp__Pow__eq,axiom,
! [A: set_li5007820469839914319term_a] :
( ( powp_l5512018315752008634term_a
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A ) )
= ( ^ [X: set_li5007820469839914319term_a] : ( member5532222276278824166term_a @ X @ ( pow_li3987379423859365436term_a @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_701_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_702_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_703_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_704_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_705_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: set_nat,L: list_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A ) ) )
= ( ( linord2614967742042102400et_nat @ A )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_706_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_707_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A ) )
= ( finite_card_nat @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_708_card__subset__eq,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( ( finite7755471413630703857term_a @ A )
= ( finite7755471413630703857term_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_709_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_710_card__subset__eq,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( ( finite6994273861720659345term_a @ A )
= ( finite6994273861720659345term_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_711_infinite__arbitrarily__large,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ~ ( finite491101672212324272term_a @ A )
=> ? [B8: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B8 )
& ( ( finite7755471413630703857term_a @ B8 )
= N )
& ( ord_le549404264473380271term_a @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_712_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_713_infinite__arbitrarily__large,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ~ ( finite8953276157157055568term_a @ A )
=> ? [B8: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B8 )
& ( ( finite6994273861720659345term_a @ B8 )
= N )
& ( ord_le6553944718276964943term_a @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_714_card__le__if__inj__on__rel,axiom,
! [B: set_li5007820469839914319term_a,A: set_fs1788988886788723183term_a,R2: fset_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( finite491101672212324272term_a @ B )
=> ( ! [A4: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A4 @ A )
=> ? [B9: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: fset_Bot_bot_term_a,A22: fset_Bot_bot_term_a,B5: list_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A1 @ A )
=> ( ( member2089387167371741008term_a @ A22 @ A )
=> ( ( member2850584719281785520term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_715_card__le__if__inj__on__rel,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a,R2: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
( ( finite491101672212324272term_a @ B )
=> ( ! [A4: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A4 @ A )
=> ? [B9: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: list_Bot_bot_term_a,A22: list_Bot_bot_term_a,B5: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A1 @ A )
=> ( ( member2850584719281785520term_a @ A22 @ A )
=> ( ( member2850584719281785520term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_716_card__le__if__inj__on__rel,axiom,
! [B: set_li5007820469839914319term_a,A: set_nat,R2: nat > list_Bot_bot_term_a > $o] :
( ( finite491101672212324272term_a @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B9: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B5: list_Bot_bot_term_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member2850584719281785520term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_717_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_fs1788988886788723183term_a,R2: fset_Bot_bot_term_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A4 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: fset_Bot_bot_term_a,A22: fset_Bot_bot_term_a,B5: nat] :
( ( member2089387167371741008term_a @ A1 @ A )
=> ( ( member2089387167371741008term_a @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_718_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_li5007820469839914319term_a,R2: list_Bot_bot_term_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A4 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: list_Bot_bot_term_a,A22: list_Bot_bot_term_a,B5: nat] :
( ( member2850584719281785520term_a @ A1 @ A )
=> ( ( member2850584719281785520term_a @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_719_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_720_card__le__if__inj__on__rel,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a,R2: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( finite8953276157157055568term_a @ B )
=> ( ! [A4: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A4 @ A )
=> ? [B9: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: fset_Bot_bot_term_a,A22: fset_Bot_bot_term_a,B5: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ A1 @ A )
=> ( ( member2089387167371741008term_a @ A22 @ A )
=> ( ( member2089387167371741008term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_721_card__le__if__inj__on__rel,axiom,
! [B: set_fs1788988886788723183term_a,A: set_li5007820469839914319term_a,R2: list_Bot_bot_term_a > fset_Bot_bot_term_a > $o] :
( ( finite8953276157157055568term_a @ B )
=> ( ! [A4: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A4 @ A )
=> ? [B9: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: list_Bot_bot_term_a,A22: list_Bot_bot_term_a,B5: fset_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ A1 @ A )
=> ( ( member2850584719281785520term_a @ A22 @ A )
=> ( ( member2089387167371741008term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_722_card__le__if__inj__on__rel,axiom,
! [B: set_fs1788988886788723183term_a,A: set_nat,R2: nat > fset_Bot_bot_term_a > $o] :
( ( finite8953276157157055568term_a @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B9: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ B9 @ B )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B5: fset_Bot_bot_term_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member2089387167371741008term_a @ B5 @ B )
=> ( ( R2 @ A1 @ B5 )
=> ( ( R2 @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_723_card__lists__length__eq,axiom,
! [A: set_li5007820469839914319term_a,N: nat] :
( ( finite491101672212324272term_a @ A )
=> ( ( finite2535492425817382017term_a
@ ( collec7320990309202468094term_a
@ ^ [Xs3: list_l7107345091518559913term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs3 ) @ A )
& ( ( size_s1109596070125842749term_a @ Xs3 )
= N ) ) ) )
= ( power_power_nat @ ( finite7755471413630703857term_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_724_card__lists__length__eq,axiom,
! [A: set_fs1788988886788723183term_a,N: nat] :
( ( finite8953276157157055568term_a @ A )
=> ( ( finite8540032879620966689term_a
@ ( collec4102158726151276958term_a
@ ^ [Xs3: list_f3888513508467368777term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs3 ) @ A )
& ( ( size_s7114136523929427421term_a @ Xs3 )
= N ) ) ) )
= ( power_power_nat @ ( finite6994273861720659345term_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_725_card__lists__length__eq,axiom,
! [A: set_Bot_bot_term_a,N: nat] :
( ( finite4019351022360529184term_a @ A )
=> ( ( finite7755471413630703857term_a
@ ( collec4020393894392429550term_a
@ ^ [Xs3: list_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs3 ) @ A )
& ( ( size_s1103687553077312429term_a @ Xs3 )
= N ) ) ) )
= ( power_power_nat @ ( finite427276431539050209term_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_726_card__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_nat @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_727_card__image__le,axiom,
! [A: set_li5007820469839914319term_a,F: list_Bot_bot_term_a > nat] :
( ( finite491101672212324272term_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_1574777119431997168_a_nat @ F @ A ) ) @ ( finite7755471413630703857term_a @ A ) ) ) ).
% card_image_le
thf(fact_728_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_729_card__image__le,axiom,
! [A: set_fs1788988886788723183term_a,F: fset_Bot_bot_term_a > nat] :
( ( finite8953276157157055568term_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_8474690510679110544_a_nat @ F @ A ) ) @ ( finite6994273861720659345term_a @ A ) ) ) ).
% card_image_le
thf(fact_730_card__mono,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ).
% card_mono
thf(fact_731_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_732_card__mono,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ).
% card_mono
thf(fact_733_card__seteq,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ B ) @ ( finite7755471413630703857term_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_734_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_735_card__seteq,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ B ) @ ( finite6994273861720659345term_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_736_exists__subset__between,axiom,
! [A: set_li5007820469839914319term_a,N: nat,C2: set_li5007820469839914319term_a] :
( ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite7755471413630703857term_a @ C2 ) )
=> ( ( ord_le549404264473380271term_a @ A @ C2 )
=> ( ( finite491101672212324272term_a @ C2 )
=> ? [B8: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ A @ B8 )
& ( ord_le549404264473380271term_a @ B8 @ C2 )
& ( ( finite7755471413630703857term_a @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_737_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_738_exists__subset__between,axiom,
! [A: set_fs1788988886788723183term_a,N: nat,C2: set_fs1788988886788723183term_a] :
( ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite6994273861720659345term_a @ C2 ) )
=> ( ( ord_le6553944718276964943term_a @ A @ C2 )
=> ( ( finite8953276157157055568term_a @ C2 )
=> ? [B8: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ A @ B8 )
& ( ord_le6553944718276964943term_a @ B8 @ C2 )
& ( ( finite6994273861720659345term_a @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_739_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_li5007820469839914319term_a] :
( ( ord_less_eq_nat @ N @ ( finite7755471413630703857term_a @ S ) )
=> ~ ! [T4: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ T4 @ S )
=> ( ( ( finite7755471413630703857term_a @ T4 )
= N )
=> ~ ( finite491101672212324272term_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_740_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_741_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_fs1788988886788723183term_a] :
( ( ord_less_eq_nat @ N @ ( finite6994273861720659345term_a @ S ) )
=> ~ ! [T4: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ T4 @ S )
=> ( ( ( finite6994273861720659345term_a @ T4 )
= N )
=> ~ ( finite8953276157157055568term_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_742_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_li5007820469839914319term_a,C2: nat] :
( ! [G3: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ G3 @ F2 )
=> ( ( finite491101672212324272term_a @ G3 )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ G3 ) @ C2 ) ) )
=> ( ( finite491101672212324272term_a @ F2 )
& ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_743_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_744_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_fs1788988886788723183term_a,C2: nat] :
( ! [G3: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ G3 @ F2 )
=> ( ( finite8953276157157055568term_a @ G3 )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ G3 ) @ C2 ) ) )
=> ( ( finite8953276157157055568term_a @ F2 )
& ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_745_card__length,axiom,
! [Xs2: list_Bot_bot_term_a] : ( ord_less_eq_nat @ ( finite427276431539050209term_a @ ( set_Bot_bot_term_a2 @ Xs2 ) ) @ ( size_s1103687553077312429term_a @ Xs2 ) ) ).
% card_length
thf(fact_746_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_747_psubset__card__mono,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le4209214868046015395term_a @ A @ B )
=> ( ord_less_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_748_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_749_psubset__card__mono,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le990383284994824259term_a @ A @ B )
=> ( ord_less_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_750_card__distinct,axiom,
! [Xs2: list_Bot_bot_term_a] :
( ( ( finite427276431539050209term_a @ ( set_Bot_bot_term_a2 @ Xs2 ) )
= ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( distin8359246687647904464term_a @ Xs2 ) ) ).
% card_distinct
thf(fact_751_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_752_distinct__card,axiom,
! [Xs2: list_Bot_bot_term_a] :
( ( distin8359246687647904464term_a @ Xs2 )
=> ( ( finite427276431539050209term_a @ ( set_Bot_bot_term_a2 @ Xs2 ) )
= ( size_s1103687553077312429term_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_753_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_754_surj__card__le,axiom,
! [A: set_li5007820469839914319term_a,B: set_nat,F: list_Bot_bot_term_a > nat] :
( ( finite491101672212324272term_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1574777119431997168_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite7755471413630703857term_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_755_surj__card__le,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 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_756_surj__card__le,axiom,
! [A: set_fs1788988886788723183term_a,B: set_nat,F: fset_Bot_bot_term_a > nat] :
( ( finite8953276157157055568term_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_8474690510679110544_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite6994273861720659345term_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_757_card__psubset,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ A @ B )
=> ( ( ord_less_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) )
=> ( ord_le4209214868046015395term_a @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_758_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_759_card__psubset,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ A @ B )
=> ( ( ord_less_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) )
=> ( ord_le990383284994824259term_a @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_760_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_761_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_762_finite__enum__subset,axiom,
! [X5: set_nat,Y7: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I2 )
= ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y7 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ ( finite_card_nat @ Y7 ) )
=> ( ord_less_eq_set_nat @ X5 @ Y7 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_763_sorted__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_764_sorted__nth__mono,axiom,
! [Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_765_enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_766_enumerate__mono__le__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_767_enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).
% enumerate_in_set
thf(fact_768_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S @ N3 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_769_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_Bot_bot_term_a,Z: list_Bot_bot_term_a] : ( Y3 = Z ) )
= ( ^ [Xs3: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs3 )
= ( size_s1103687553077312429term_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ( ( nth_Bot_bot_term_a @ Xs3 @ I4 )
= ( nth_Bot_bot_term_a @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_770_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_771_Skolem__list__nth,axiom,
! [K: nat,P: nat > bot_bot_term_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: bot_bot_term_a] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_Bot_bot_term_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_772_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: nat] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_773_nth__equalityI,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_s1103687553077312429term_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ Xs2 @ I2 )
= ( nth_Bot_bot_term_a @ Ys @ I2 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_774_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_775_Ex__list__of__length__P,axiom,
! [N: nat,P: bot_bot_term_a > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: bot_bot_term_a] : ( P @ X6 @ I2 ) )
=> ? [Xs: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs )
= N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( nth_Bot_bot_term_a @ Xs @ I3 ) @ I3 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_776_Ex__list__of__length__P,axiom,
! [N: nat,P: nat > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: nat] : ( P @ X6 @ I2 ) )
=> ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( nth_nat @ Xs @ I3 ) @ I3 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_777_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_778_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_779_all__set__conv__all__nth,axiom,
! [Xs2: list_Bot_bot_term_a,P: bot_bot_term_a > $o] :
( ( ! [X: bot_bot_term_a] :
( ( member2723211829317350432term_a @ X @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_780_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_781_all__nth__imp__all__set,axiom,
! [Xs2: list_f3888513508467368777term_a,P: fset_Bot_bot_term_a > $o,X2: fset_Bot_bot_term_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7114136523929427421term_a @ Xs2 ) )
=> ( P @ ( nth_fs1711276827724382834term_a @ Xs2 @ I2 ) ) )
=> ( ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_782_all__nth__imp__all__set,axiom,
! [Xs2: list_l7107345091518559913term_a,P: list_Bot_bot_term_a > $o,X2: list_Bot_bot_term_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1109596070125842749term_a @ Xs2 ) )
=> ( P @ ( nth_li2472474379634427346term_a @ Xs2 @ I2 ) ) )
=> ( ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_783_all__nth__imp__all__set,axiom,
! [Xs2: list_Bot_bot_term_a,P: bot_bot_term_a > $o,X2: bot_bot_term_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ Xs2 @ I2 ) ) )
=> ( ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_784_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_785_in__set__conv__nth,axiom,
! [X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s7114136523929427421term_a @ Xs2 ) )
& ( ( nth_fs1711276827724382834term_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_786_in__set__conv__nth,axiom,
! [X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1109596070125842749term_a @ Xs2 ) )
& ( ( nth_li2472474379634427346term_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_787_in__set__conv__nth,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_788_in__set__conv__nth,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_789_list__ball__nth,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a,P: bot_bot_term_a > $o] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ! [X3: bot_bot_term_a] :
( ( member2723211829317350432term_a @ X3 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_Bot_bot_term_a @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_790_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_791_nth__mem,axiom,
! [N: nat,Xs2: list_f3888513508467368777term_a] :
( ( ord_less_nat @ N @ ( size_s7114136523929427421term_a @ Xs2 ) )
=> ( member2089387167371741008term_a @ ( nth_fs1711276827724382834term_a @ Xs2 @ N ) @ ( set_fs4478461793750015460term_a @ Xs2 ) ) ) ).
% nth_mem
thf(fact_792_nth__mem,axiom,
! [N: nat,Xs2: list_l7107345091518559913term_a] :
( ( ord_less_nat @ N @ ( size_s1109596070125842749term_a @ Xs2 ) )
=> ( member2850584719281785520term_a @ ( nth_li2472474379634427346term_a @ Xs2 @ N ) @ ( set_li5239659345660059972term_a @ Xs2 ) ) ) ).
% nth_mem
thf(fact_793_nth__mem,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( member2723211829317350432term_a @ ( nth_Bot_bot_term_a @ Xs2 @ N ) @ ( set_Bot_bot_term_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_794_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_795_ex__set__conv__ex__nth,axiom,
! [Xs2: list_Bot_bot_term_a,P: bot_bot_term_a > $o] :
( ( ? [X: bot_bot_term_a] :
( ( member2723211829317350432term_a @ X @ ( set_Bot_bot_term_a2 @ Xs2 ) )
& ( P @ X ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( P @ ( nth_Bot_bot_term_a @ Xs2 @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_796_ex__set__conv__ex__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( P @ X ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_797_list__all2__conv__all__nth,axiom,
( list_a6841061213177516955term_a
= ( ^ [P3: bot_bot_term_a > bot_bot_term_a > $o,Xs3: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs3 )
= ( size_s1103687553077312429term_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ( P3 @ ( nth_Bot_bot_term_a @ Xs3 @ I4 ) @ ( nth_Bot_bot_term_a @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_798_list__all2__conv__all__nth,axiom,
( list_a782220476339921762_a_nat
= ( ^ [P3: bot_bot_term_a > nat > $o,Xs3: list_Bot_bot_term_a,Ys2: list_nat] :
( ( ( size_s1103687553077312429term_a @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ( P3 @ ( nth_Bot_bot_term_a @ Xs3 @ I4 ) @ ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_799_list__all2__conv__all__nth,axiom,
( list_a8962304646142687458term_a
= ( ^ [P3: nat > bot_bot_term_a > $o,Xs3: list_nat,Ys2: list_Bot_bot_term_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s1103687553077312429term_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_Bot_bot_term_a @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_800_list__all2__conv__all__nth,axiom,
( list_all2_nat_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_801_list__all2__all__nthI,axiom,
! [A2: list_Bot_bot_term_a,B3: list_Bot_bot_term_a,P: bot_bot_term_a > bot_bot_term_a > $o] :
( ( ( size_s1103687553077312429term_a @ A2 )
= ( size_s1103687553077312429term_a @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1103687553077312429term_a @ A2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ A2 @ N3 ) @ ( nth_Bot_bot_term_a @ B3 @ N3 ) ) )
=> ( list_a6841061213177516955term_a @ P @ A2 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_802_list__all2__all__nthI,axiom,
! [A2: list_Bot_bot_term_a,B3: list_nat,P: bot_bot_term_a > nat > $o] :
( ( ( size_s1103687553077312429term_a @ A2 )
= ( size_size_list_nat @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1103687553077312429term_a @ A2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ A2 @ N3 ) @ ( nth_nat @ B3 @ N3 ) ) )
=> ( list_a782220476339921762_a_nat @ P @ A2 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_803_list__all2__all__nthI,axiom,
! [A2: list_nat,B3: list_Bot_bot_term_a,P: nat > bot_bot_term_a > $o] :
( ( ( size_size_list_nat @ A2 )
= ( size_s1103687553077312429term_a @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ A2 ) )
=> ( P @ ( nth_nat @ A2 @ N3 ) @ ( nth_Bot_bot_term_a @ B3 @ N3 ) ) )
=> ( list_a8962304646142687458term_a @ P @ A2 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_804_list__all2__all__nthI,axiom,
! [A2: list_nat,B3: list_nat,P: nat > nat > $o] :
( ( ( size_size_list_nat @ A2 )
= ( size_size_list_nat @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ A2 ) )
=> ( P @ ( nth_nat @ A2 @ N3 ) @ ( nth_nat @ B3 @ N3 ) ) )
=> ( list_all2_nat_nat @ P @ A2 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_805_list__all2__nthD2,axiom,
! [P: nat > bot_bot_term_a > $o,Xs2: list_nat,Ys: list_Bot_bot_term_a,P5: nat] :
( ( list_a8962304646142687458term_a @ P @ Xs2 @ Ys )
=> ( ( ord_less_nat @ P5 @ ( size_s1103687553077312429term_a @ Ys ) )
=> ( P @ ( nth_nat @ Xs2 @ P5 ) @ ( nth_Bot_bot_term_a @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD2
thf(fact_806_list__all2__nthD2,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,P5: nat] :
( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
=> ( ( ord_less_nat @ P5 @ ( size_size_list_nat @ Ys ) )
=> ( P @ ( nth_nat @ Xs2 @ P5 ) @ ( nth_nat @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD2
thf(fact_807_list__all2__nthD,axiom,
! [P: bot_bot_term_a > nat > $o,Xs2: list_Bot_bot_term_a,Ys: list_nat,P5: nat] :
( ( list_a782220476339921762_a_nat @ P @ Xs2 @ Ys )
=> ( ( ord_less_nat @ P5 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ Xs2 @ P5 ) @ ( nth_nat @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD
thf(fact_808_list__all2__nthD,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,P5: nat] :
( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
=> ( ( ord_less_nat @ P5 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ P5 ) @ ( nth_nat @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD
thf(fact_809_distinct__conv__nth,axiom,
( distin8359246687647904464term_a
= ( ^ [Xs3: list_Bot_bot_term_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth_Bot_bot_term_a @ Xs3 @ I4 )
!= ( nth_Bot_bot_term_a @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_810_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth_nat @ Xs3 @ I4 )
!= ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_811_nth__eq__iff__index__eq,axiom,
! [Xs2: list_Bot_bot_term_a,I: nat,J: nat] :
( ( distin8359246687647904464term_a @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ( nth_Bot_bot_term_a @ Xs2 @ I )
= ( nth_Bot_bot_term_a @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_812_nth__eq__iff__index__eq,axiom,
! [Xs2: list_nat,I: nat,J: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_813_sorted__wrt__iff__nth__less,axiom,
( sorted156305998458754396term_a
= ( ^ [P3: bot_bot_term_a > bot_bot_term_a > $o,Xs3: list_Bot_bot_term_a] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_s1103687553077312429term_a @ Xs3 ) )
=> ( P3 @ ( nth_Bot_bot_term_a @ Xs3 @ I4 ) @ ( nth_Bot_bot_term_a @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_814_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_815_sorted__wrt__nth__less,axiom,
! [P: bot_bot_term_a > bot_bot_term_a > $o,Xs2: list_Bot_bot_term_a,I: nat,J: nat] :
( ( sorted156305998458754396term_a @ P @ Xs2 )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ Xs2 @ I ) @ ( nth_Bot_bot_term_a @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_816_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs2 )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_817_enumerate__mono,axiom,
! [M: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M @ N )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% enumerate_mono
thf(fact_818_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_819_finite__enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_820_finite__enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N3 )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_821_finite__enum__ext,axiom,
! [X5: set_nat,Y7: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I2 )
= ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y7 )
=> ( ( ( finite_card_nat @ X5 )
= ( finite_card_nat @ Y7 ) )
=> ( X5 = Y7 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_822_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_823_distinct__Ex1,axiom,
! [Xs2: list_f3888513508467368777term_a,X2: fset_Bot_bot_term_a] :
( ( distin7553967534715678208term_a @ Xs2 )
=> ( ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s7114136523929427421term_a @ Xs2 ) )
& ( ( nth_fs1711276827724382834term_a @ Xs2 @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s7114136523929427421term_a @ Xs2 ) )
& ( ( nth_fs1711276827724382834term_a @ Xs2 @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_824_distinct__Ex1,axiom,
! [Xs2: list_l7107345091518559913term_a,X2: list_Bot_bot_term_a] :
( ( distin8315165086625722720term_a @ Xs2 )
=> ( ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s1109596070125842749term_a @ Xs2 ) )
& ( ( nth_li2472474379634427346term_a @ Xs2 @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s1109596070125842749term_a @ Xs2 ) )
& ( ( nth_li2472474379634427346term_a @ Xs2 @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_825_distinct__Ex1,axiom,
! [Xs2: list_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( distin8359246687647904464term_a @ Xs2 )
=> ( ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_826_distinct__Ex1,axiom,
! [Xs2: list_nat,X2: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_827_finite__enumerate__mono,axiom,
! [M: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M @ N )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_828_inv__to__set,axiom,
! [Ss: list_f3888513508467368777term_a,S: set_fs1788988886788723183term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s7114136523929427421term_a @ Ss ) )
=> ( member2089387167371741008term_a @ ( nth_fs1711276827724382834term_a @ Ss @ I4 ) @ S ) ) )
= ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Ss ) @ S ) ) ).
% inv_to_set
thf(fact_829_inv__to__set,axiom,
! [Ss: list_l7107345091518559913term_a,S: set_li5007820469839914319term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1109596070125842749term_a @ Ss ) )
=> ( member2850584719281785520term_a @ ( nth_li2472474379634427346term_a @ Ss @ I4 ) @ S ) ) )
= ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Ss ) @ S ) ) ).
% inv_to_set
thf(fact_830_inv__to__set,axiom,
! [Ss: list_Bot_bot_term_a,S: set_Bot_bot_term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ss ) )
=> ( member2723211829317350432term_a @ ( nth_Bot_bot_term_a @ Ss @ I4 ) @ S ) ) )
= ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Ss ) @ S ) ) ).
% inv_to_set
thf(fact_831_inv__to__set,axiom,
! [Ss: list_nat,S: set_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I4 ) @ S ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ S ) ) ).
% inv_to_set
thf(fact_832_subseteq__set__conv__nth,axiom,
! [Ss: list_f3888513508467368777term_a,T: set_fs1788988886788723183term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s7114136523929427421term_a @ Ss ) )
=> ( member2089387167371741008term_a @ ( nth_fs1711276827724382834term_a @ Ss @ I4 ) @ T ) ) )
= ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Ss ) @ T ) ) ).
% subseteq_set_conv_nth
thf(fact_833_subseteq__set__conv__nth,axiom,
! [Ss: list_l7107345091518559913term_a,T: set_li5007820469839914319term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1109596070125842749term_a @ Ss ) )
=> ( member2850584719281785520term_a @ ( nth_li2472474379634427346term_a @ Ss @ I4 ) @ T ) ) )
= ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Ss ) @ T ) ) ).
% subseteq_set_conv_nth
thf(fact_834_subseteq__set__conv__nth,axiom,
! [Ss: list_Bot_bot_term_a,T: set_Bot_bot_term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ss ) )
=> ( member2723211829317350432term_a @ ( nth_Bot_bot_term_a @ Ss @ I4 ) @ T ) ) )
= ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Ss ) @ T ) ) ).
% subseteq_set_conv_nth
thf(fact_835_subseteq__set__conv__nth,axiom,
! [Ss: list_nat,T: set_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I4 ) @ T ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T ) ) ).
% subseteq_set_conv_nth
thf(fact_836_set__list__subset__eq__nth__conv,axiom,
! [Xs2: list_f3888513508467368777term_a,A: set_fs1788988886788723183term_a] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s7114136523929427421term_a @ Xs2 ) )
=> ( member2089387167371741008term_a @ ( nth_fs1711276827724382834term_a @ Xs2 @ I4 ) @ A ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_837_set__list__subset__eq__nth__conv,axiom,
! [Xs2: list_l7107345091518559913term_a,A: set_li5007820469839914319term_a] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1109596070125842749term_a @ Xs2 ) )
=> ( member2850584719281785520term_a @ ( nth_li2472474379634427346term_a @ Xs2 @ I4 ) @ A ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_838_set__list__subset__eq__nth__conv,axiom,
! [Xs2: list_Bot_bot_term_a,A: set_Bot_bot_term_a] :
( ( ord_le6342072807802007839term_a @ ( set_Bot_bot_term_a2 @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( member2723211829317350432term_a @ ( nth_Bot_bot_term_a @ Xs2 @ I4 ) @ A ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_839_set__list__subset__eq__nth__conv,axiom,
! [Xs2: list_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ I4 ) @ A ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_840_in__set__idx,axiom,
! [X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7114136523929427421term_a @ Xs2 ) )
& ( ( nth_fs1711276827724382834term_a @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_set_idx
thf(fact_841_in__set__idx,axiom,
! [X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1109596070125842749term_a @ Xs2 ) )
& ( ( nth_li2472474379634427346term_a @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_set_idx
thf(fact_842_in__set__idx,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_set_idx
thf(fact_843_in__set__idx,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_set_idx
thf(fact_844_nth__equalityE,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ( Xs2 = Ys )
=> ~ ( ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_s1103687553077312429term_a @ Ys ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ Xs2 @ I3 )
= ( nth_Bot_bot_term_a @ Ys @ I3 ) ) ) ) ) ).
% nth_equalityE
thf(fact_845_nth__equalityE,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ~ ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) ) ) ) ).
% nth_equalityE
thf(fact_846_permut__sound,axiom,
! [I: nat,As2: list_Bot_bot_term_a,F: nat > nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ As2 ) )
=> ( ( nth_Bot_bot_term_a @ ( missin150706566486609201term_a @ As2 @ F ) @ I )
= ( nth_Bot_bot_term_a @ As2 @ ( F @ I ) ) ) ) ).
% permut_sound
thf(fact_847_permut__sound,axiom,
! [I: nat,As2: list_nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As2 ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As2 @ F ) @ I )
= ( nth_nat @ As2 @ ( F @ I ) ) ) ) ).
% permut_sound
thf(fact_848_permut__aux__sound,axiom,
! [I: nat,As2: list_Bot_bot_term_a,F: nat > nat,Bs: list_Bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ As2 ) )
=> ( ( nth_Bot_bot_term_a @ ( missin828916799783925063term_a @ As2 @ F @ Bs ) @ I )
= ( nth_Bot_bot_term_a @ Bs @ ( F @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_849_permut__aux__sound,axiom,
! [I: nat,As2: list_nat,F: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As2 ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As2 @ F @ Bs ) @ I )
= ( nth_nat @ Bs @ ( F @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_850_min__list__nth,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_851_sorted__rev__nth__mono,axiom,
! [Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_852_length__rev,axiom,
! [Xs2: list_Bot_bot_term_a] :
( ( size_s1103687553077312429term_a @ ( rev_Bot_bot_term_a @ Xs2 ) )
= ( size_s1103687553077312429term_a @ Xs2 ) ) ).
% length_rev
thf(fact_853_length__rev,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rev
thf(fact_854_fset__of__list__rev,axiom,
! [Xs2: list_Bot_bot_term_a] :
( ( fset_o7715858782369100227term_a @ ( rev_Bot_bot_term_a @ Xs2 ) )
= ( fset_o7715858782369100227term_a @ Xs2 ) ) ).
% fset_of_list_rev
thf(fact_855_sorted__wrt__rev,axiom,
! [P: nat > nat > $o,Xs2: list_nat] :
( ( sorted_wrt_nat @ P @ ( rev_nat @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y4: nat] : ( P @ Y4 @ X )
@ Xs2 ) ) ).
% sorted_wrt_rev
thf(fact_856_min__list,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ X2 ) ) ).
% min_list
thf(fact_857_min__list__subset,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ ( missing_min_list_nat @ Xs2 ) @ ( set_nat2 @ Ys ) )
=> ( ( missing_min_list_nat @ Xs2 )
= ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_subset
thf(fact_858_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J3 ) @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_859_sorted__rev__iff__nth__Suc,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ ( suc @ I4 ) ) @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_860_finite__enumerate__step,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_861_sorted__iff__nth__Suc,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_862_fset__list__fsubset__eq__nth__conv,axiom,
! [Xs2: list_nat,A: fset_nat] :
( ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( fmember_nat @ ( nth_nat @ Xs2 @ I4 ) @ A ) ) ) ) ).
% fset_list_fsubset_eq_nth_conv
thf(fact_863_fset__list__fsubset__eq__nth__conv,axiom,
! [Xs2: list_Bot_bot_term_a,A: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( fset_o7715858782369100227term_a @ Xs2 ) @ A )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( fmembe1418995468851910914term_a @ ( nth_Bot_bot_term_a @ Xs2 @ I4 ) @ A ) ) ) ) ).
% fset_list_fsubset_eq_nth_conv
thf(fact_864_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_865_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_866_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_867_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_868_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_869_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_870_fsubsetI,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ! [X3: bot_bot_term_a] :
( ( fmembe1418995468851910914term_a @ X3 @ A )
=> ( fmembe1418995468851910914term_a @ X3 @ B ) )
=> ( ord_le7216997114146882585term_a @ A @ B ) ) ).
% fsubsetI
thf(fact_871_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_872_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_873_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_874_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_875_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_876_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_877_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_878_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_879_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_880_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_881_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_882_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_883_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_884_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_885_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_886_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_887_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_888_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_889_fin__mono,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( fmembe1418995468851910914term_a @ X2 @ A )
=> ( fmembe1418995468851910914term_a @ X2 @ B ) ) ) ).
% fin_mono
thf(fact_890_fsubsetD,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C: bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( fmembe1418995468851910914term_a @ C @ A )
=> ( fmembe1418995468851910914term_a @ C @ B ) ) ) ).
% fsubsetD
thf(fact_891_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_892_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_893_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_894_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_895_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_896_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_897_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_898_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_899_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_900_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_901_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_902_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_903_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_904_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_905_lift__Suc__antimono__le,axiom,
! [F: nat > fset_Bot_bot_term_a,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_le7216997114146882585term_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_le7216997114146882585term_a @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_906_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_907_lift__Suc__mono__le,axiom,
! [F: nat > fset_Bot_bot_term_a,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_le7216997114146882585term_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_le7216997114146882585term_a @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_908_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_909_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_910_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_911_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_912_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_913_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_914_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_915_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_916_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_917_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_918_fset__of__list__elem,axiom,
! [X2: nat,Xs2: list_nat] :
( ( fmember_nat @ X2 @ ( fset_of_list_nat @ Xs2 ) )
= ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% fset_of_list_elem
thf(fact_919_fset__of__list__elem,axiom,
! [X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( fmembe1129996945036628786term_a @ X2 @ ( fset_o7938989536108156787term_a @ Xs2 ) )
= ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) ) ) ).
% fset_of_list_elem
thf(fact_920_fset__of__list__elem,axiom,
! [X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( fmembe1891194496946673298term_a @ X2 @ ( fset_o8700187088018201299term_a @ Xs2 ) )
= ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) ) ) ).
% fset_of_list_elem
thf(fact_921_fset__of__list__elem,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( fmembe1418995468851910914term_a @ X2 @ ( fset_o7715858782369100227term_a @ Xs2 ) )
= ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) ) ) ).
% fset_of_list_elem
thf(fact_922_enumerate__step,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).
% enumerate_step
thf(fact_923_in__fset__idx,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( fmembe1418995468851910914term_a @ X2 @ ( fset_o7715858782369100227term_a @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_fset_idx
thf(fact_924_in__fset__idx,axiom,
! [X2: nat,Xs2: list_nat] :
( ( fmember_nat @ X2 @ ( fset_of_list_nat @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_fset_idx
thf(fact_925_in__fset__conv__nth,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( fmembe1418995468851910914term_a @ X2 @ ( fset_o7715858782369100227term_a @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Xs2 ) )
& ( ( nth_Bot_bot_term_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_fset_conv_nth
thf(fact_926_in__fset__conv__nth,axiom,
! [X2: nat,Xs2: list_nat] :
( ( fmember_nat @ X2 @ ( fset_of_list_nat @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_fset_conv_nth
thf(fact_927_fnth__mem,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( fmembe1418995468851910914term_a @ ( nth_Bot_bot_term_a @ Xs2 @ N ) @ ( fset_o7715858782369100227term_a @ Xs2 ) ) ) ).
% fnth_mem
thf(fact_928_fnth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( fmember_nat @ ( nth_nat @ Xs2 @ N ) @ ( fset_of_list_nat @ Xs2 ) ) ) ).
% fnth_mem
thf(fact_929_fsubseteq__fset__conv__nth,axiom,
! [Ss: list_nat,T: fset_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( fmember_nat @ ( nth_nat @ Ss @ I4 ) @ T ) ) )
= ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Ss ) @ T ) ) ).
% fsubseteq_fset_conv_nth
thf(fact_930_fsubseteq__fset__conv__nth,axiom,
! [Ss: list_Bot_bot_term_a,T: fset_Bot_bot_term_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ss ) )
=> ( fmembe1418995468851910914term_a @ ( nth_Bot_bot_term_a @ Ss @ I4 ) @ T ) ) )
= ( ord_le7216997114146882585term_a @ ( fset_o7715858782369100227term_a @ Ss ) @ T ) ) ).
% fsubseteq_fset_conv_nth
thf(fact_931_finite__enumerate__Suc_H_H,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
=> ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
= ( ord_Least_nat
@ ^ [S3: nat] :
( ( member_nat @ S3 @ S )
& ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S3 ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_932_remove__nth__sound__r,axiom,
! [N: nat,P5: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_eq_nat @ N @ P5 )
=> ( ( ord_less_nat @ P5 @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ N @ Xs2 ) @ P5 )
= ( nth_Bot_bot_term_a @ Xs2 @ ( suc @ P5 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_933_remove__nth__sound__r,axiom,
! [N: nat,P5: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ P5 )
=> ( ( ord_less_nat @ P5 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P5 )
= ( nth_nat @ Xs2 @ ( suc @ P5 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_934_LeastI2__wellorder__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [A4: nat] :
( ( P @ A4 )
=> ( ! [B9: nat] :
( ( P @ B9 )
=> ( ord_less_eq_nat @ A4 @ B9 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_935_LeastI2__wellorder,axiom,
! [P: nat > $o,A2: nat,Q: nat > $o] :
( ( P @ A2 )
=> ( ! [A4: nat] :
( ( P @ A4 )
=> ( ! [B9: nat] :
( ( P @ B9 )
=> ( ord_less_eq_nat @ A4 @ B9 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder
thf(fact_936_Least__equality,axiom,
! [P: fset_Bot_bot_term_a > $o,X2: fset_Bot_bot_term_a] :
( ( P @ X2 )
=> ( ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ X2 @ Y2 ) )
=> ( ( ord_Le6783622042733668521term_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_937_Least__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( ord_Least_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_938_LeastI2__order,axiom,
! [P: fset_Bot_bot_term_a > $o,X2: fset_Bot_bot_term_a,Q: fset_Bot_bot_term_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ X2 @ Y2 ) )
=> ( ! [X3: fset_Bot_bot_term_a] :
( ( P @ X3 )
=> ( ! [Y5: fset_Bot_bot_term_a] :
( ( P @ Y5 )
=> ( ord_le7216997114146882585term_a @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Le6783622042733668521term_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_939_LeastI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_940_Least1__le,axiom,
! [P: fset_Bot_bot_term_a > $o,Z2: fset_Bot_bot_term_a] :
( ? [X6: fset_Bot_bot_term_a] :
( ( P @ X6 )
& ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ X6 @ Y2 ) )
& ! [Y2: fset_Bot_bot_term_a] :
( ( ( P @ Y2 )
& ! [Ya2: fset_Bot_bot_term_a] :
( ( P @ Ya2 )
=> ( ord_le7216997114146882585term_a @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z2 )
=> ( ord_le7216997114146882585term_a @ ( ord_Le6783622042733668521term_a @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_941_Least1__le,axiom,
! [P: nat > $o,Z2: nat] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya2: nat] :
( ( P @ Ya2 )
=> ( ord_less_eq_nat @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_942_Least1I,axiom,
! [P: fset_Bot_bot_term_a > $o] :
( ? [X6: fset_Bot_bot_term_a] :
( ( P @ X6 )
& ! [Y2: fset_Bot_bot_term_a] :
( ( P @ Y2 )
=> ( ord_le7216997114146882585term_a @ X6 @ Y2 ) )
& ! [Y2: fset_Bot_bot_term_a] :
( ( ( P @ Y2 )
& ! [Ya2: fset_Bot_bot_term_a] :
( ( P @ Ya2 )
=> ( ord_le7216997114146882585term_a @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Le6783622042733668521term_a @ P ) ) ) ).
% Least1I
thf(fact_943_Least1I,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya2: nat] :
( ( P @ Ya2 )
=> ( ord_less_eq_nat @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% Least1I
thf(fact_944_LeastI2,axiom,
! [P: nat > $o,A2: nat,Q: nat > $o] :
( ( P @ A2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2
thf(fact_945_LeastI__ex,axiom,
! [P: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% LeastI_ex
thf(fact_946_LeastI2__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_ex
thf(fact_947_LeastI,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% LeastI
thf(fact_948_Least__le,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).
% Least_le
thf(fact_949_not__less__Least,axiom,
! [K: nat,P: nat > $o] :
( ( ord_less_nat @ K @ ( ord_Least_nat @ P ) )
=> ~ ( P @ K ) ) ).
% not_less_Least
thf(fact_950_remove__nth__id,axiom,
! [Xs2: list_Bot_bot_term_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ N )
=> ( ( missin3275086306548970576term_a @ N @ Xs2 )
= Xs2 ) ) ).
% remove_nth_id
thf(fact_951_remove__nth__id,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs2 )
= Xs2 ) ) ).
% remove_nth_id
thf(fact_952_remove__nth__sound__l,axiom,
! [P5: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ P5 @ N )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P5 )
= ( nth_nat @ Xs2 @ P5 ) ) ) ).
% remove_nth_sound_l
thf(fact_953_remove__nth__len,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( size_s1103687553077312429term_a @ Xs2 )
= ( suc @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ I @ Xs2 ) ) ) ) ) ).
% remove_nth_len
thf(fact_954_remove__nth__len,axiom,
! [I: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ).
% remove_nth_len
thf(fact_955_remove__nth__P__compat,axiom,
! [As2: list_Bot_bot_term_a,Bs: list_Bot_bot_term_a,P: bot_bot_term_a > bot_bot_term_a > $o,P5: nat] :
( ( ( size_s1103687553077312429term_a @ As2 )
= ( size_s1103687553077312429term_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ As2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ As2 @ I2 ) @ ( nth_Bot_bot_term_a @ Bs @ I2 ) ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ P5 @ As2 ) ) )
=> ( P @ ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ P5 @ As2 ) @ I3 ) @ ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ P5 @ Bs ) @ I3 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_956_remove__nth__P__compat,axiom,
! [As2: list_Bot_bot_term_a,Bs: list_nat,P: bot_bot_term_a > nat > $o,P5: nat] :
( ( ( size_s1103687553077312429term_a @ As2 )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ As2 ) )
=> ( P @ ( nth_Bot_bot_term_a @ As2 @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ P5 @ As2 ) ) )
=> ( P @ ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ P5 @ As2 ) @ I3 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ Bs ) @ I3 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_957_remove__nth__P__compat,axiom,
! [As2: list_nat,Bs: list_Bot_bot_term_a,P: nat > bot_bot_term_a > $o,P5: nat] :
( ( ( size_size_list_nat @ As2 )
= ( size_s1103687553077312429term_a @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As2 ) )
=> ( P @ ( nth_nat @ As2 @ I2 ) @ ( nth_Bot_bot_term_a @ Bs @ I2 ) ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P5 @ As2 ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ As2 ) @ I3 ) @ ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ P5 @ Bs ) @ I3 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_958_remove__nth__P__compat,axiom,
! [As2: list_nat,Bs: list_nat,P: nat > nat > $o,P5: nat] :
( ( ( size_size_list_nat @ As2 )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As2 ) )
=> ( P @ ( nth_nat @ As2 @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P5 @ As2 ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ As2 ) @ I3 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P5 @ Bs ) @ I3 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_959_enumerate__Suc_H_H,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
= ( ord_Least_nat
@ ^ [S3: nat] :
( ( member_nat @ S3 @ S )
& ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S3 ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_960_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ N @ Xs2 ) ) )
=> ( ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ N @ Xs2 ) @ I )
= ( nth_Bot_bot_term_a @ Xs2 @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_961_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ I )
= ( nth_nat @ Xs2 @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_962_adjust__idx__rev__nth,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( J != I )
=> ( ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ I @ Xs2 ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_Bot_bot_term_a @ Xs2 @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_963_adjust__idx__rev__nth,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( J != I )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_nat @ Xs2 @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_964_adjust__idx__nth,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ ( missin3275086306548970576term_a @ I @ Xs2 ) @ J )
= ( nth_Bot_bot_term_a @ Xs2 @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_965_adjust__idx__nth,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ J )
= ( nth_nat @ Xs2 @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_966_adjust__idx__rev__length,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ I @ Xs2 ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_967_adjust__idx__rev__length,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_968_adjust__idx__i,axiom,
! [I: nat,J: nat] :
( ( missing_adjust_idx @ I @ J )
!= I ) ).
% adjust_idx_i
thf(fact_969_adjust__idx__rev2,axiom,
! [J: nat,I: nat] :
( ( J != I )
=> ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_970_adjust__idx__rev1,axiom,
! [I: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_971_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( suc @ J3 ) ) ) ) ).
% adjust_idx_def
thf(fact_972_adjust__idx__length,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ I @ Xs2 ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_s1103687553077312429term_a @ Xs2 ) ) ) ) ).
% adjust_idx_length
thf(fact_973_adjust__idx__length,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% adjust_idx_length
thf(fact_974_abort__Bleast__def,axiom,
( abort_1540119917928410727term_a
= ( ^ [S4: set_fs1788988886788723183term_a,P3: fset_Bot_bot_term_a > $o] :
( ord_Le6783622042733668521term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ S4 )
& ( P3 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_975_abort__Bleast__def,axiom,
( abort_Bleast_nat
= ( ^ [S4: set_nat,P3: nat > $o] :
( ord_Least_nat
@ ^ [X: nat] :
( ( member_nat @ X @ S4 )
& ( P3 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_976_Bleast__def,axiom,
( bleast229174303562731211term_a
= ( ^ [S4: set_fs1788988886788723183term_a,P3: fset_Bot_bot_term_a > $o] :
( ord_Le6783622042733668521term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ S4 )
& ( P3 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_977_Bleast__def,axiom,
( bleast_nat
= ( ^ [S4: set_nat,P3: nat > $o] :
( ord_Least_nat
@ ^ [X: nat] :
( ( member_nat @ X @ S4 )
& ( P3 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_978_length__add__elem__list__lists,axiom,
! [Ys: list_Bot_bot_term_a,X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Ys @ ( set_li5239659345660059972term_a @ ( basic_8661219279717973152term_a @ X2 @ Xs2 ) ) )
=> ( ( size_s1103687553077312429term_a @ Ys )
= ( suc @ ( size_s1103687553077312429term_a @ Xs2 ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_979_length__add__elem__list__lists,axiom,
! [Ys: list_nat,X2: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X2 @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_980_rev__nth,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ ( rev_Bot_bot_term_a @ Xs2 ) @ N )
= ( nth_Bot_bot_term_a @ Xs2 @ ( minus_minus_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_981_rev__nth,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( rev_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_982_trancl__listp_Obase,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a,R3: bot_bot_term_a > bot_bot_term_a > $o] :
( ( ( size_s1103687553077312429term_a @ Xs2 )
= ( size_s1103687553077312429term_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Ys ) )
=> ( R3 @ ( nth_Bot_bot_term_a @ Xs2 @ I2 ) @ ( nth_Bot_bot_term_a @ Ys @ I2 ) ) )
=> ( trancl4041116690992654595term_a @ R3 @ Xs2 @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_983_trancl__listp_Obase,axiom,
! [Xs2: list_nat,Ys: list_nat,R3: nat > nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( R3 @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( trancl_listp_nat @ R3 @ Xs2 @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_984_distinct__swap,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( distin8359246687647904464term_a @ ( list_u5503441972953478683term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ ( nth_Bot_bot_term_a @ Xs2 @ J ) ) @ J @ ( nth_Bot_bot_term_a @ Xs2 @ I ) ) )
= ( distin8359246687647904464term_a @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_985_distinct__swap,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
= ( distinct_nat @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_986_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_987_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_988_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_989_length__list__update,axiom,
! [Xs2: list_Bot_bot_term_a,I: nat,X2: bot_bot_term_a] :
( ( size_s1103687553077312429term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 ) )
= ( size_s1103687553077312429term_a @ Xs2 ) ) ).
% length_list_update
thf(fact_990_length__list__update,axiom,
! [Xs2: list_nat,I: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_991_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs2: list_nat,X2: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_992_list__update__id,axiom,
! [Xs2: list_nat,I: nat] :
( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_993_list__update__beyond,axiom,
! [Xs2: list_Bot_bot_term_a,I: nat,X2: bot_bot_term_a] :
( ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ I )
=> ( ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_994_list__update__beyond,axiom,
! [Xs2: list_nat,I: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
=> ( ( list_update_nat @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_995_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( nth_Bot_bot_term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_996_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_997_set__swap,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( set_Bot_bot_term_a2 @ ( list_u5503441972953478683term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ ( nth_Bot_bot_term_a @ Xs2 @ J ) ) @ J @ ( nth_Bot_bot_term_a @ Xs2 @ I ) ) )
= ( set_Bot_bot_term_a2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_998_set__swap,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_999_trancl__listp_Ocases,axiom,
! [R3: bot_bot_term_a > bot_bot_term_a > $o,A12: list_Bot_bot_term_a,A23: list_Bot_bot_term_a] :
( ( trancl4041116690992654595term_a @ R3 @ A12 @ A23 )
=> ( ( ( ( size_s1103687553077312429term_a @ A12 )
= ( size_s1103687553077312429term_a @ A23 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ A23 ) )
=> ( R3 @ ( nth_Bot_bot_term_a @ A12 @ I3 ) @ ( nth_Bot_bot_term_a @ A23 @ I3 ) ) ) )
=> ~ ! [Ys4: list_Bot_bot_term_a,I2: nat,Z4: bot_bot_term_a] :
( ( A23
= ( list_u5503441972953478683term_a @ Ys4 @ I2 @ Z4 ) )
=> ( ( trancl4041116690992654595term_a @ R3 @ A12 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Ys4 ) )
=> ~ ( R3 @ ( nth_Bot_bot_term_a @ Ys4 @ I2 ) @ Z4 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_1000_trancl__listp_Ocases,axiom,
! [R3: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( trancl_listp_nat @ R3 @ A12 @ A23 )
=> ( ( ( ( size_size_list_nat @ A12 )
= ( size_size_list_nat @ A23 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ A23 ) )
=> ( R3 @ ( nth_nat @ A12 @ I3 ) @ ( nth_nat @ A23 @ I3 ) ) ) )
=> ~ ! [Ys4: list_nat,I2: nat,Z4: nat] :
( ( A23
= ( list_update_nat @ Ys4 @ I2 @ Z4 ) )
=> ( ( trancl_listp_nat @ R3 @ A12 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( R3 @ ( nth_nat @ Ys4 @ I2 ) @ Z4 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_1001_trancl__listp_Osimps,axiom,
( trancl4041116690992654595term_a
= ( ^ [R4: bot_bot_term_a > bot_bot_term_a > $o,A13: list_Bot_bot_term_a,A24: list_Bot_bot_term_a] :
( ? [Xs3: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
( ( A13 = Xs3 )
& ( A24 = Ys2 )
& ( ( size_s1103687553077312429term_a @ Xs3 )
= ( size_s1103687553077312429term_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ys2 ) )
=> ( R4 @ ( nth_Bot_bot_term_a @ Xs3 @ I4 ) @ ( nth_Bot_bot_term_a @ Ys2 @ I4 ) ) ) )
| ? [Xs3: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a,I4: nat,Z6: bot_bot_term_a] :
( ( A13 = Xs3 )
& ( A24
= ( list_u5503441972953478683term_a @ Ys2 @ I4 @ Z6 ) )
& ( trancl4041116690992654595term_a @ R4 @ Xs3 @ Ys2 )
& ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ys2 ) )
& ( R4 @ ( nth_Bot_bot_term_a @ Ys2 @ I4 ) @ Z6 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_1002_trancl__listp_Osimps,axiom,
( trancl_listp_nat
= ( ^ [R4: nat > nat > $o,A13: list_nat,A24: list_nat] :
( ? [Xs3: list_nat,Ys2: list_nat] :
( ( A13 = Xs3 )
& ( A24 = Ys2 )
& ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
=> ( R4 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Ys2 @ I4 ) ) ) )
| ? [Xs3: list_nat,Ys2: list_nat,I4: nat,Z6: nat] :
( ( A13 = Xs3 )
& ( A24
= ( list_update_nat @ Ys2 @ I4 @ Z6 ) )
& ( trancl_listp_nat @ R4 @ Xs3 @ Ys2 )
& ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
& ( R4 @ ( nth_nat @ Ys2 @ I4 ) @ Z6 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_1003_trancl__listp_Olist__trancl,axiom,
! [R3: bot_bot_term_a > bot_bot_term_a > $o,Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a,I: nat,Z2: bot_bot_term_a] :
( ( trancl4041116690992654595term_a @ R3 @ Xs2 @ Ys )
=> ( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Ys ) )
=> ( ( R3 @ ( nth_Bot_bot_term_a @ Ys @ I ) @ Z2 )
=> ( trancl4041116690992654595term_a @ R3 @ Xs2 @ ( list_u5503441972953478683term_a @ Ys @ I @ Z2 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_1004_trancl__listp_Olist__trancl,axiom,
! [R3: nat > nat > $o,Xs2: list_nat,Ys: list_nat,I: nat,Z2: nat] :
( ( trancl_listp_nat @ R3 @ Xs2 @ Ys )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( R3 @ ( nth_nat @ Ys @ I ) @ Z2 )
=> ( trancl_listp_nat @ R3 @ Xs2 @ ( list_update_nat @ Ys @ I @ Z2 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_1005_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1006_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1007_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1008_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1009_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1010_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1011_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1012_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1013_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1014_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1015_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1016_set__update__subsetI,axiom,
! [Xs2: list_nat,A: set_nat,X2: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1017_set__update__subsetI,axiom,
! [Xs2: list_f3888513508467368777term_a,A: set_fs1788988886788723183term_a,X2: fset_Bot_bot_term_a,I: nat] :
( ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ Xs2 ) @ A )
=> ( ( member2089387167371741008term_a @ X2 @ A )
=> ( ord_le6553944718276964943term_a @ ( set_fs4478461793750015460term_a @ ( list_u7318593726414639051term_a @ Xs2 @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1018_set__update__subsetI,axiom,
! [Xs2: list_l7107345091518559913term_a,A: set_li5007820469839914319term_a,X2: list_Bot_bot_term_a,I: nat] :
( ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ Xs2 ) @ A )
=> ( ( member2850584719281785520term_a @ X2 @ A )
=> ( ord_le549404264473380271term_a @ ( set_li5239659345660059972term_a @ ( list_u8079791278324683563term_a @ Xs2 @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1019_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1020_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1021_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1022_diff__less__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1023_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1024_set__update__memI,axiom,
! [N: nat,Xs2: list_f3888513508467368777term_a,X2: fset_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s7114136523929427421term_a @ Xs2 ) )
=> ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ ( list_u7318593726414639051term_a @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1025_set__update__memI,axiom,
! [N: nat,Xs2: list_l7107345091518559913term_a,X2: list_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1109596070125842749term_a @ Xs2 ) )
=> ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ ( list_u8079791278324683563term_a @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1026_set__update__memI,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ ( list_u5503441972953478683term_a @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1027_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1028_list__update__same__conv,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 )
= Xs2 )
= ( ( nth_Bot_bot_term_a @ Xs2 @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1029_list__update__same__conv,axiom,
! [I: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I @ X2 )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1030_nth__list__update,axiom,
! [I: nat,Xs2: list_Bot_bot_term_a,J: nat,X2: bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth_Bot_bot_term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_Bot_bot_term_a @ ( list_u5503441972953478683term_a @ Xs2 @ I @ X2 ) @ J )
= ( nth_Bot_bot_term_a @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1031_nth__list__update,axiom,
! [I: nat,Xs2: list_nat,J: nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1032_parallel__list__update,axiom,
! [N: nat,R2: bot_bot_term_a > bot_bot_term_a > $o,P5: list_Bot_bot_term_a > $o,Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
( ! [Xs: list_Bot_bot_term_a,I2: nat,Y2: bot_bot_term_a] :
( ( ( size_s1103687553077312429term_a @ Xs )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_Bot_bot_term_a @ Xs @ I2 ) @ Y2 )
=> ( ( P5 @ Xs )
=> ( P5 @ ( list_u5503441972953478683term_a @ Xs @ I2 @ Y2 ) ) ) ) ) )
=> ( ( ( size_s1103687553077312429term_a @ Xs2 )
= N )
=> ( ( P5 @ Xs2 )
=> ( ( ( size_s1103687553077312429term_a @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_Bot_bot_term_a @ Xs2 @ I2 ) @ ( nth_Bot_bot_term_a @ Ys @ I2 ) ) )
=> ( P5 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_1033_parallel__list__update,axiom,
! [N: nat,R2: nat > nat > $o,P5: list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ! [Xs: list_nat,I2: nat,Y2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_nat @ Xs @ I2 ) @ Y2 )
=> ( ( P5 @ Xs )
=> ( P5 @ ( list_update_nat @ Xs @ I2 @ Y2 ) ) ) ) ) )
=> ( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ( P5 @ Xs2 )
=> ( ( ( size_size_list_nat @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( P5 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_1034_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_1035_Diff__iff,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( minus_1840015707743988168term_a @ A @ B ) )
= ( ( member2089387167371741008term_a @ C @ A )
& ~ ( member2089387167371741008term_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1036_Diff__iff,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( minus_5058847290795179304term_a @ A @ B ) )
= ( ( member2850584719281785520term_a @ C @ A )
& ~ ( member2850584719281785520term_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1037_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_1038_DiffI,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ A )
=> ( ~ ( member2089387167371741008term_a @ C @ B )
=> ( member2089387167371741008term_a @ C @ ( minus_1840015707743988168term_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1039_DiffI,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ A )
=> ( ~ ( member2850584719281785520term_a @ C @ B )
=> ( member2850584719281785520term_a @ C @ ( minus_5058847290795179304term_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1040_finite__Diff,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ( finite491101672212324272term_a @ ( minus_5058847290795179304term_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1041_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1042_finite__Diff,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( finite8953276157157055568term_a @ ( minus_1840015707743988168term_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1043_finite__Diff2,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( finite491101672212324272term_a @ ( minus_5058847290795179304term_a @ A @ B ) )
= ( finite491101672212324272term_a @ A ) ) ) ).
% finite_Diff2
thf(fact_1044_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1045_finite__Diff2,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( finite8953276157157055568term_a @ ( minus_1840015707743988168term_a @ A @ B ) )
= ( finite8953276157157055568term_a @ A ) ) ) ).
% finite_Diff2
thf(fact_1046_fminus__mono,axiom,
! [A: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a,D: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ C2 )
=> ( ( ord_le7216997114146882585term_a @ D @ B )
=> ( ord_le7216997114146882585term_a @ ( minus_7937957780725882770term_a @ A @ B ) @ ( minus_7937957780725882770term_a @ C2 @ D ) ) ) ) ).
% fminus_mono
thf(fact_1047_double__fminus,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( ord_le7216997114146882585term_a @ B @ C2 )
=> ( ( minus_7937957780725882770term_a @ B @ ( minus_7937957780725882770term_a @ C2 @ A ) )
= A ) ) ) ).
% double_fminus
thf(fact_1048_fminus__fsubset,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] : ( ord_le7216997114146882585term_a @ ( minus_7937957780725882770term_a @ A @ B ) @ A ) ).
% fminus_fsubset
thf(fact_1049_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1050_psubset__imp__ex__mem,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( ord_le990383284994824259term_a @ A @ B )
=> ? [B5: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ B5 @ ( minus_1840015707743988168term_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1051_psubset__imp__ex__mem,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( ord_le4209214868046015395term_a @ A @ B )
=> ? [B5: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ B5 @ ( minus_5058847290795179304term_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1052_set__diff__eq,axiom,
( minus_1840015707743988168term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( collec3259196342482385038term_a
@ ^ [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A5 )
& ~ ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1053_set__diff__eq,axiom,
( minus_5058847290795179304term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( collec4020393894392429550term_a
@ ^ [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A5 )
& ~ ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1054_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A5 )
& ~ ( member_nat @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1055_minus__set__def,axiom,
( minus_1840015707743988168term_a
= ( ^ [A5: set_fs1788988886788723183term_a,B2: set_fs1788988886788723183term_a] :
( collec3259196342482385038term_a
@ ( minus_8274016294857714365rm_a_o
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ A5 )
@ ^ [X: fset_Bot_bot_term_a] : ( member2089387167371741008term_a @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1056_minus__set__def,axiom,
( minus_5058847290795179304term_a
= ( ^ [A5: set_li5007820469839914319term_a,B2: set_li5007820469839914319term_a] :
( collec4020393894392429550term_a
@ ( minus_1564044971597811549rm_a_o
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ A5 )
@ ^ [X: list_Bot_bot_term_a] : ( member2850584719281785520term_a @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1057_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1058_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_1059_DiffD2,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( minus_1840015707743988168term_a @ A @ B ) )
=> ~ ( member2089387167371741008term_a @ C @ B ) ) ).
% DiffD2
thf(fact_1060_DiffD2,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( minus_5058847290795179304term_a @ A @ B ) )
=> ~ ( member2850584719281785520term_a @ C @ B ) ) ).
% DiffD2
thf(fact_1061_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_1062_DiffD1,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( minus_1840015707743988168term_a @ A @ B ) )
=> ( member2089387167371741008term_a @ C @ A ) ) ).
% DiffD1
thf(fact_1063_DiffD1,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( minus_5058847290795179304term_a @ A @ B ) )
=> ( member2850584719281785520term_a @ C @ A ) ) ).
% DiffD1
thf(fact_1064_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_1065_DiffE,axiom,
! [C: fset_Bot_bot_term_a,A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( member2089387167371741008term_a @ C @ ( minus_1840015707743988168term_a @ A @ B ) )
=> ~ ( ( member2089387167371741008term_a @ C @ A )
=> ( member2089387167371741008term_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1066_DiffE,axiom,
! [C: list_Bot_bot_term_a,A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( member2850584719281785520term_a @ C @ ( minus_5058847290795179304term_a @ A @ B ) )
=> ~ ( ( member2850584719281785520term_a @ C @ A )
=> ( member2850584719281785520term_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1067_Diff__infinite__finite,axiom,
! [T: set_li5007820469839914319term_a,S: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ T )
=> ( ~ ( finite491101672212324272term_a @ S )
=> ~ ( finite491101672212324272term_a @ ( minus_5058847290795179304term_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1068_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1069_Diff__infinite__finite,axiom,
! [T: set_fs1788988886788723183term_a,S: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ T )
=> ( ~ ( finite8953276157157055568term_a @ S )
=> ~ ( finite8953276157157055568term_a @ ( minus_1840015707743988168term_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1070_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1071_fminus__fsubset__conv,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a,C2: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( minus_7937957780725882770term_a @ A @ B ) @ C2 )
= ( ord_le7216997114146882585term_a @ A @ ( sup_su6230847479081161957term_a @ B @ C2 ) ) ) ).
% fminus_fsubset_conv
thf(fact_1072_fminus__partition,axiom,
! [A: fset_Bot_bot_term_a,B: fset_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ A @ B )
=> ( ( sup_su6230847479081161957term_a @ A @ ( minus_7937957780725882770term_a @ B @ A ) )
= B ) ) ).
% fminus_partition
thf(fact_1073_card__less__sym__Diff,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ( ( finite491101672212324272term_a @ B )
=> ( ( ord_less_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) )
=> ( ord_less_nat @ ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ A @ B ) ) @ ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1074_card__less__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1075_card__less__sym__Diff,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( finite8953276157157055568term_a @ B )
=> ( ( ord_less_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) )
=> ( ord_less_nat @ ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ A @ B ) ) @ ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1076_card__le__sym__Diff,axiom,
! [A: set_li5007820469839914319term_a,B: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ( ( finite491101672212324272term_a @ B )
=> ( ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) )
=> ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ A @ B ) ) @ ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1077_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1078_card__le__sym__Diff,axiom,
! [A: set_fs1788988886788723183term_a,B: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( finite8953276157157055568term_a @ B )
=> ( ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) )
=> ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ A @ B ) ) @ ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1079_card__Diff__subset,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ( ord_le549404264473380271term_a @ B @ A )
=> ( ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ A @ B ) )
= ( minus_minus_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1080_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1081_card__Diff__subset,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ( ord_le6553944718276964943term_a @ B @ A )
=> ( ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ A @ B ) )
= ( minus_minus_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1082_diff__card__le__card__Diff,axiom,
! [B: set_li5007820469839914319term_a,A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite7755471413630703857term_a @ A ) @ ( finite7755471413630703857term_a @ B ) ) @ ( finite7755471413630703857term_a @ ( minus_5058847290795179304term_a @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1083_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1084_diff__card__le__card__Diff,axiom,
! [B: set_fs1788988886788723183term_a,A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite6994273861720659345term_a @ A ) @ ( finite6994273861720659345term_a @ B ) ) @ ( finite6994273861720659345term_a @ ( minus_1840015707743988168term_a @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1085_set__subtract__list__sorted,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( set_nat2 @ ( missin6424796737333596952ed_nat @ Xs2 @ Ys ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ) ) ).
% set_subtract_list_sorted
thf(fact_1086_rev__update,axiom,
! [K: nat,Xs2: list_Bot_bot_term_a,Y: bot_bot_term_a] :
( ( ord_less_nat @ K @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( rev_Bot_bot_term_a @ ( list_u5503441972953478683term_a @ Xs2 @ K @ Y ) )
= ( list_u5503441972953478683term_a @ ( rev_Bot_bot_term_a @ Xs2 ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1087_rev__update,axiom,
! [K: nat,Xs2: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs2 @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs2 ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1088_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1089_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1090_sorted__wrt01,axiom,
! [Xs2: list_Bot_bot_term_a,P: bot_bot_term_a > bot_bot_term_a > $o] :
( ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ one_one_nat )
=> ( sorted156305998458754396term_a @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_1091_sorted__wrt01,axiom,
! [Xs2: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_1092_sorted01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted01
thf(fact_1093_remove__nth__length,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( ord_less_nat @ N @ ( size_s1103687553077312429term_a @ Xs2 ) )
=> ( ( size_s1103687553077312429term_a @ ( missin3275086306548970576term_a @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_1094_remove__nth__length,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_1095_power__increasing__iff,axiom,
! [B3: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1096_power__strict__increasing__iff,axiom,
! [B3: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_nat @ X2 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1097_power__inject__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M )
= ( power_power_nat @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1098_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1099_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_1100_one__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_1101_power__gt1,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1102_power__strict__increasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1103_power__less__imp__less__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1104_power__increasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1105_power__le__imp__le__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1106_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1107_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1108_power__decreasing__iff,axiom,
! [B3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1109_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1110_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_1111_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1112_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1113_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1114_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1115_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1116_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1117_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1118_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1119_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1120_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1121_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1122_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1123_power__Suc0__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_1124_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1125_card_Oinfinite,axiom,
! [A: set_li5007820469839914319term_a] :
( ~ ( finite491101672212324272term_a @ A )
=> ( ( finite7755471413630703857term_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1126_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1127_card_Oinfinite,axiom,
! [A: set_fs1788988886788723183term_a] :
( ~ ( finite8953276157157055568term_a @ A )
=> ( ( finite6994273861720659345term_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1128_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1129_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1130_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1131_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1132_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1133_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1134_power__eq__0__iff,axiom,
! [A2: nat,N: nat] :
( ( ( power_power_nat @ A2 @ N )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1135_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1136_power__mono__iff,axiom,
! [A2: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1137_power__strict__decreasing__iff,axiom,
! [B3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1138_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1139_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1140_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1141_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1142_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1143_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1144_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1145_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1146_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1147_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1148_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1149_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1150_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1151_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1152_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1153_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1154_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1155_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1156_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1157_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1158_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1159_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1160_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1161_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1162_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1163_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1164_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1165_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1166_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1167_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1168_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1169_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1170_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1171_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1172_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1173_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1174_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1175_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1176_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1177_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1178_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1179_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1180_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1181_power__strict__mono,axiom,
! [A2: nat,B3: nat,N: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1182_power__mono,axiom,
! [A2: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ).
% power_mono
thf(fact_1183_zero__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_1184_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1185_power__eq__imp__eq__base,axiom,
! [A2: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A2 @ N )
= ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A2 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1186_power__eq__iff__eq__base,axiom,
! [N: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ( power_power_nat @ A2 @ N )
= ( power_power_nat @ B3 @ N ) )
= ( A2 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1187_power__not__zero,axiom,
! [A2: nat,N: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_1188_zero__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_less_power
thf(fact_1189_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1190_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1191_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1192_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1193_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1194_enumerate__0,axiom,
! [S: set_nat] :
( ( infini8530281810654367211te_nat @ S @ zero_zero_nat )
= ( ord_Least_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ S ) ) ) ).
% enumerate_0
thf(fact_1195_power__less__imp__less__base,axiom,
! [A2: nat,N: nat,B3: nat] :
( ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_1196_power__le__one,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1197_power__inject__base,axiom,
! [A2: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_1198_power__le__imp__le__base,axiom,
! [A2: nat,N: nat,B3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_1199_length__pos__if__in__set,axiom,
! [X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s7114136523929427421term_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1200_length__pos__if__in__set,axiom,
! [X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1109596070125842749term_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1201_length__pos__if__in__set,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2723211829317350432term_a @ X2 @ ( set_Bot_bot_term_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1103687553077312429term_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1202_length__pos__if__in__set,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1203_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1204_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1205_card__ge__0__finite,axiom,
! [A: set_li5007820469839914319term_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite7755471413630703857term_a @ A ) )
=> ( finite491101672212324272term_a @ A ) ) ).
% card_ge_0_finite
thf(fact_1206_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_1207_card__ge__0__finite,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6994273861720659345term_a @ A ) )
=> ( finite8953276157157055568term_a @ A ) ) ).
% card_ge_0_finite
thf(fact_1208_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1209_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1210_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1211_card__less,axiom,
! [M3: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M3 )
& ( ord_less_nat @ K5 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1212_card__less__Suc,axiom,
! [M3: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M3 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ ( suc @ K5 ) @ M3 )
& ( ord_less_nat @ K5 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M3 )
& ( ord_less_nat @ K5 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1213_card__less__Suc2,axiom,
! [M3: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ ( suc @ K5 ) @ M3 )
& ( ord_less_nat @ K5 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M3 )
& ( ord_less_nat @ K5 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1214_power__Suc__le__self,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_1215_power__Suc__less__one,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1216_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1217_power__decreasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1218_self__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% self_le_power
thf(fact_1219_one__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% one_less_power
thf(fact_1220_card__le__Suc0__iff__eq,axiom,
! [A: set_li5007820469839914319term_a] :
( ( finite491101672212324272term_a @ A )
=> ( ( ord_less_eq_nat @ ( finite7755471413630703857term_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ X @ A )
=> ! [Y4: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Y4 @ A )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1221_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1222_card__le__Suc0__iff__eq,axiom,
! [A: set_fs1788988886788723183term_a] :
( ( finite8953276157157055568term_a @ A )
=> ( ( ord_less_eq_nat @ ( finite6994273861720659345term_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ X @ A )
=> ! [Y4: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Y4 @ A )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1223_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1224_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1225_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1226_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1227_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1228_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1229_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1230_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1231_last__nthI,axiom,
! [I: nat,Ts: list_Bot_bot_term_a] :
( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Ts ) )
=> ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_s1103687553077312429term_a @ Ts ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_Bot_bot_term_a @ Ts @ I )
= ( last_Bot_bot_term_a @ Ts ) ) ) ) ).
% last_nthI
thf(fact_1232_last__nthI,axiom,
! [I: nat,Ts: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
=> ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_size_list_nat @ Ts ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_nat @ Ts @ I )
= ( last_nat @ Ts ) ) ) ) ).
% last_nthI
thf(fact_1233_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1234_nth__Cons__Suc,axiom,
! [X2: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_1235_nth__Cons__0,axiom,
! [X2: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_1236_impossible__Cons,axiom,
! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a,X2: bot_bot_term_a] :
( ( ord_less_eq_nat @ ( size_s1103687553077312429term_a @ Xs2 ) @ ( size_s1103687553077312429term_a @ Ys ) )
=> ( Xs2
!= ( cons_Bot_bot_term_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1237_impossible__Cons,axiom,
! [Xs2: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs2
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1238_Suc__length__conv,axiom,
! [N: nat,Xs2: list_Bot_bot_term_a] :
( ( ( suc @ N )
= ( size_s1103687553077312429term_a @ Xs2 ) )
= ( ? [Y4: bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
( ( Xs2
= ( cons_Bot_bot_term_a @ Y4 @ Ys2 ) )
& ( ( size_s1103687553077312429term_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1239_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1240_length__Suc__conv,axiom,
! [Xs2: list_Bot_bot_term_a,N: nat] :
( ( ( size_s1103687553077312429term_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
( ( Xs2
= ( cons_Bot_bot_term_a @ Y4 @ Ys2 ) )
& ( ( size_s1103687553077312429term_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1241_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ Y4 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1242_length__Cons,axiom,
! [X2: bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( size_s1103687553077312429term_a @ ( cons_Bot_bot_term_a @ X2 @ Xs2 ) )
= ( suc @ ( size_s1103687553077312429term_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_1243_length__Cons,axiom,
! [X2: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_1244_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1245_set__ConsD,axiom,
! [Y: fset_Bot_bot_term_a,X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ ( cons_f4155457494393317507term_a @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1246_set__ConsD,axiom,
! [Y: list_Bot_bot_term_a,X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ ( cons_l4916655046303362019term_a @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1247_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1248_list_Oset__cases,axiom,
! [E: fset_Bot_bot_term_a,A2: list_f3888513508467368777term_a] :
( ( member2089387167371741008term_a @ E @ ( set_fs4478461793750015460term_a @ A2 ) )
=> ( ! [Z22: list_f3888513508467368777term_a] :
( A2
!= ( cons_f4155457494393317507term_a @ E @ Z22 ) )
=> ~ ! [Z1: fset_Bot_bot_term_a,Z22: list_f3888513508467368777term_a] :
( ( A2
= ( cons_f4155457494393317507term_a @ Z1 @ Z22 ) )
=> ~ ( member2089387167371741008term_a @ E @ ( set_fs4478461793750015460term_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1249_list_Oset__cases,axiom,
! [E: list_Bot_bot_term_a,A2: list_l7107345091518559913term_a] :
( ( member2850584719281785520term_a @ E @ ( set_li5239659345660059972term_a @ A2 ) )
=> ( ! [Z22: list_l7107345091518559913term_a] :
( A2
!= ( cons_l4916655046303362019term_a @ E @ Z22 ) )
=> ~ ! [Z1: list_Bot_bot_term_a,Z22: list_l7107345091518559913term_a] :
( ( A2
= ( cons_l4916655046303362019term_a @ Z1 @ Z22 ) )
=> ~ ( member2850584719281785520term_a @ E @ ( set_li5239659345660059972term_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1250_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1251_list_Oset__intros_I1_J,axiom,
! [X21: fset_Bot_bot_term_a,X222: list_f3888513508467368777term_a] : ( member2089387167371741008term_a @ X21 @ ( set_fs4478461793750015460term_a @ ( cons_f4155457494393317507term_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1252_list_Oset__intros_I1_J,axiom,
! [X21: list_Bot_bot_term_a,X222: list_l7107345091518559913term_a] : ( member2850584719281785520term_a @ X21 @ ( set_li5239659345660059972term_a @ ( cons_l4916655046303362019term_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1253_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1254_list_Oset__intros_I2_J,axiom,
! [Y: fset_Bot_bot_term_a,X222: list_f3888513508467368777term_a,X21: fset_Bot_bot_term_a] :
( ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ X222 ) )
=> ( member2089387167371741008term_a @ Y @ ( set_fs4478461793750015460term_a @ ( cons_f4155457494393317507term_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1255_list_Oset__intros_I2_J,axiom,
! [Y: list_Bot_bot_term_a,X222: list_l7107345091518559913term_a,X21: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ X222 ) )
=> ( member2850584719281785520term_a @ Y @ ( set_li5239659345660059972term_a @ ( cons_l4916655046303362019term_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1256_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs2 ) )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1257_distinct_Osimps_I2_J,axiom,
! [X2: fset_Bot_bot_term_a,Xs2: list_f3888513508467368777term_a] :
( ( distin7553967534715678208term_a @ ( cons_f4155457494393317507term_a @ X2 @ Xs2 ) )
= ( ~ ( member2089387167371741008term_a @ X2 @ ( set_fs4478461793750015460term_a @ Xs2 ) )
& ( distin7553967534715678208term_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1258_distinct_Osimps_I2_J,axiom,
! [X2: list_Bot_bot_term_a,Xs2: list_l7107345091518559913term_a] :
( ( distin8315165086625722720term_a @ ( cons_l4916655046303362019term_a @ X2 @ Xs2 ) )
= ( ~ ( member2850584719281785520term_a @ X2 @ ( set_li5239659345660059972term_a @ Xs2 ) )
& ( distin8315165086625722720term_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1259_sorted2,axiom,
! [X2: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1260_Cons__in__subseqsD,axiom,
! [Y: bot_bot_term_a,Ys: list_Bot_bot_term_a,Xs2: list_Bot_bot_term_a] :
( ( member2850584719281785520term_a @ ( cons_Bot_bot_term_a @ Y @ Ys ) @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) )
=> ( member2850584719281785520term_a @ Ys @ ( set_li5239659345660059972term_a @ ( subseq3012724568006980340term_a @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1261_subtract__list__sorted_Osimps_I1_J,axiom,
! [X2: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( X2 = Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
= ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) ) )
& ( ( X2 != Y )
=> ( ( ( ord_less_nat @ X2 @ Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ X2 @ ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
= ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ Ys ) ) ) ) ) ) ).
% subtract_list_sorted.simps(1)
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( finite491101672212324272term_a
@ ( collec4020393894392429550term_a
@ ^ [Qs: list_Bot_bot_term_a] :
( ( ord_le7216997114146882585term_a @ ( fset_o7715858782369100227term_a @ Qs ) @ ( fstates_a_b @ r ) )
& ( ( size_s1103687553077312429term_a @ Qs )
= n ) ) ) ) ).
%------------------------------------------------------------------------------